in this video we're going to focus on pre-algebra we're going to cover some common topics that you might see in this course so the first thing that you need to be able to do is add and subtract integers for example let's say if we want to add 5 plus three now many of you know that five plus three is eight but if you ever have difficulty with this type of math use the number line let's start with five you could place it anywhere whenever you're adding a number to another number you need to move to the right of the number line anytime you're adding and if you need to subtract travel towards the left so in this case we want to add three to five so we need to travel three units to the right one two three this is six seven eight so therefore five plus three is eight now let's try some more examples what is negative 4 plus 5 let's use the number line to get the answer so we're going to start with negative 4 and we're going to add 5 to it so we're going to travel 5 units to the right one two three four five this is negative three negative two negative one zero one so negative four plus five is positive one let's work on another example seven minus five so we're going to start with seven and this time we're subtracting it by five so we need to travel five units to the left one two three four five so this is six five four three two so seven minus five is two now what about negative four minus two so let's start with negative four and we're going to subtract it by two so we need to go two units to the left and actually that should be negative five negative three is on the right side so this is negative six therefore negative four minus two is equal to negative six now what about this one negative six minus negative three if you were to see something that looks like that what would you do whenever you have two negative signs right next to each other it's equivalent to a positive sign when you multiply a negative by a negative it's equal to a positive number so we're looking for negative six plus three so for adding we need to travel to the right one two three this is negative five negative four negative three and so negative six plus three is negative three now what about eight plus negative five what's the answer for this one so if we start at eight and we're subtracting by five by the way this expression is equal to eight minus five a positive times a negative is a negative sign so we need to travel five units to the left this is going to be seven six five four three so eight minus five is positive three now let's talk about multiplication what is eight times three so you could answer this question easily if you have memorized your multiplication tables but in the event that you don't know just remember multiplication is simply repeated addition eight times three means that you're adding eight three times it's also equivalent to adding three eight times but it's easier to add eight three times eight plus eight is sixteen and sixteen plus eight is twenty-four so therefore eight times three is twenty-four let's work on another example what's nine times four nine times four is equivalent to adding nine four times nine plus nine is eighteen so these two nines add up to eighteen and the other two nines add up to eighteen as well and eighteen plus eighteen is thirty six so therefore nine times four is thirty six now what is negative five multiplied by three a negative times a positive number will give you a negative result so we could just focus on adding five three times and then make the entire thing negative five plus five is ten and ten plus five is fifteen so therefore negative five times three is negative fifteen try this one what is negative six multiplied by negative eight you multiply two negative numbers you're going to get a positive result so this is equivalent to multiplying six times eight so i'm going to add eight six times instead of adding six eight times now adding 2 8 will give me 16 so i have 16 plus 16 plus 16 and 16 plus 16 is 32 and 32 plus 16 is 48 so therefore negative 6 times negative 8 is equal to this number positive 48 now let's move on to our next example let's focus on division what is 54 divided by 6 now it's important to understand that division is the opposite of multiplication 6 multiplied by what number is equal to 54. so how many times do you have to add 6 to get to 54 it turns out that 6 times 9 is 54 so 54 divided by 6 is 9. so division is simply the opposite of multiplication so here's another example what is negative 45 divided by positive nine a negative number divided by a positive number will give you a negative result so we know the overall answer is negative so let's just focus on dividing 45 by nine so nine times what number is equal to negative forty five it turns out that you have to add nine five times to get to forty five nine plus nine is eighteen 18 plus 9 is 27 27 plus 9 is 36 36 plus 9 is 45 so therefore 9 times negative 5 is negative 45 and if we focus on the reverse statement negative 45 divided by 9 that's going to be negative 5. and so that's a quick and simple way to perform simple division here's another example what's negative 12 divided by negative two when you divide two negative numbers you're going to get a positive result so this is equivalent to dividing twelve by two so two times what number is twelve you have to add two six times to get to twelve two plus two is four if you add another two that's six and then eight and then ten and then twelve so therefore two times six is equal to twelve and twelve divided by two has to be six now let's say if you have this problem what is eight minus five times four so what is the answer now there's two possible ways of attempting to do this problem and one of the two ways i'm going to show you is the right answer the other is not so should we subtract first or should we multiply first if we subtract 8 minus 5 is 3 and 3 times 4 is 12. we're going to get that result but now let's say if we multiply first negative 5 times 4 is negative 20 so this becomes 8 minus 20 and 8 minus 20 is negative twelve so the results are different so which one comes first subtraction or multiplication perhaps you heard of pemdas please excuse my dear aunt sally p stands for parentheses e exponents m multiplication d division a is addition s is subtraction and so anytime you need to figure out which operation comes first look at this expression this is associated with the order of operations and parentheses have the highest priority now we're comparing multiplication and subtraction so therefore you should always multiply first before you subtract multiplication has more priority than subtraction so that's how you can use pemdas to know which operation should come first so therefore this is the correct answer eight minus five times four is negative 12. now you can confirm your answer using a scientific calculator if you have access to it simply type this expression exactly the way you see it and the answer that you should get is negative 12. now let's move on to another example try this one what is six plus 24 divided by four so feel free to take a minute and work on this example so according to pemdas division has more priority over addition so p e m d a s so as you look at the letters towards the left they have more priority over the letters on the right so d is to the left of a so division has more priority than addition so you should divide first before you add so what is 24 divided by 4 24 divided by 4 is 6 because 4 times 6 is 24 and 6 plus 6 is 12. so that's the final answer in this example now let's try another one what is eight minus five multiplied by seven so should we subtract or should we multiply first in this case in this case you should subtract you need to perform the operation inside the parenthesis so you're comparing parentheses to multiplication and you need to work inside the parentheses before you multiply so 8 minus 5 is 3 and 3 times 7 is 21 so that's the answer in this particular example now what about this problem what is 24 divided by four multiplied by three should we perform division first or multiplication now according to the word pemdas it appears that multiplication has more priority than division because it's on the left but it turns out that these two terms multiplication and division they have the same priority and addition and subtraction also have the same priority now when you see a problem like this where you can multiply or divide first you need to travel from left to right that means you should work on the operations on the left and then save the operations on the right for last so we're going to do it two ways let's divide first 24 divided by four is six six times three is eighteen now let's do it the other way let's perform multiplication first four times three is twelve and twenty 24 divided by 12 is 2. so as we could see we get different answers here if you type this in your calculator hopefully you have a scientific calculator it will give you 18 as the answer so whenever you have division and multiplication simply work from the left side to the right side and that will give you the right answer now what about a problem that looks like this in this case what should we do according to pemdas parentheses has more priority than multiplication and division so in this case we need to work inside the parentheses 4 times 3 is 12 and so we have 24 divided by 12 which is 2. and if you type this in exactly the way you see it in a scientific calculator you should get two as your answer and that's how you could confirm all of these problems just type it in the calculator and see what you get now let's work on another problem what is 48 divided by eight minus two multiply by three so first we need to work inside the parenthesis eight minus two is six so we have 48 divided by six times three now that we have division and multiplication we need to work starting from the left towards the right 48 divided by 6 is 8 and 8 times 3 is 24 and so that's going to be the final answer for this problem here's another example what is 32 minus 24 divided by 8 divided by 2. so feel free to pause the video and simplify this expression 8 divided by 2 is 4 and 32 minus 24 is 8 and 8 divided by 4 is 2. so that's going to be the final answer in this example try these two problems what's 7 multiplied by 9 minus 4 and what is 3 times 4 plus 8 minus 2 divided by 2. so the one above is simple we need to work inside the parenthesis first nine minus four is five and seven times five is thirty five now let's work on this example so first we need to subtract eight by two eight minus two is six and now we need to work inside the brackets six divided by two that's equal to three so we have three four plus three now what's our next step four plus three is seven and 3 times 7 is 21 so that's the final answer for that example now sometimes you may need to evaluate algebraic expressions for example let's say if we have the expression x y divided by two plus five and let's say that you're told x is equal to four and y is equal to three what is the value of this expression if you see a question like this all you need to do is replace x with its value x is equal to four and y we're going to replace it with three so what we now have is four times three divided by two plus five four times three is twelve and twelve divided by two is six six plus five is 11. so that is the value of this expression given x equals 4 and y equals 3. let's work on another example evaluate the expression let's say the expression is 4x plus 3y minus 2z and let's say that x is equal to 5 y is 2 and z is equal to 3. so all we need to do is substitute we need to replace x with its value of 5 and we're going to replace y with 2 and z with 3 and then just perform the operation 4 times 5 is 20. 3 times 2 is 6 two times three is also six six minus six is zero and twenty plus zero is simply twenty so therefore that's the value of this expression let's try another example what is 5x minus 2 times y plus z so let's say x is 3 y is 7 and z is 4. so feel free to pause the video and evaluate this expression so let's replace x with three and y with seven and z is four so don't forget to perform order of operations we need to add seven plus four we could multiply five times three simultaneously that's going to be fifteen seven times four is eleven now we need to multiply before we subtract two times 11 is 22. so what we have is 15 minus 22 which will give you negative 7 and so that's the end result for this problem try this one x squared minus y squared divided by four z plus eight and let's say that x is equal to eight y is six and let's say z is four so x squared will be replaced with eight squared and y let's replace it with six and then let's substitute z with four so this is the expression that we need to simplify so now what is 8 squared 8 squared is 8 times 8 which is 64. 6 squared or 6 times 6 that's equal to 36 and on the bottom we have 4 times 4 which is 16. now what is 64 minus 36 if we use a calculator that's equal to 28 and 16 plus 8 is 24. now this fraction is reducible so how can we reduce this improper fraction 28 is seven times four twenty-four is six times four four divided by four is one so we can cancel it so what we have left over is seven divided by six and that is the answer what would you do if you saw an expression that looks like this what is 3 times x plus 4 so we can't really add x plus 4. x is a variable in which we don't know or have a value for so we can't evaluate the expression but we can simplify it so how can we do so now there's something called the distributive property we need to distribute three to x and four three multiplied by x is simply three x and three multiplied by positive four is twelve so this expression is equal to 3x plus 12. let's try another example now what is 4 multiplied by 2x minus 3 go ahead and use the distributive property 4 multiplied by 2x is equal to 8x and 4 multiplied to negative 3 is negative 12. so this is equal to 8x minus 12. now sometimes you may have some other algebraic expressions to simplify here's another one what is 5x plus 3x go ahead and simplify all you need to do is add the coefficients 5 plus 3 is 8 so this is equal to 8x now what about this what's 7y plus 2y plus 8. so based on the last example go ahead and simplify this expression what we need to do is add like terms 7y plus 2y and that's equal to 9y now we cannot add 9y and 8 because what is it going to be 17 or 17y because the 8 doesn't have a y it's not a similar term to 9y so we cannot add them therefore the final answer is 9y plus 8. try this one 3 times x plus 5 added to 8x now before we could do anything we need to perform or use the distributive property so we got to distribute 3 to x which we know it's going to be 3x and we have to multiply 3 and 5 which is 15. now the only common terms that we have are 3x and 8x they're similar they both carry the variable x three plus eight is eleven so three x plus eight x is eleven x therefore the final answer is eleven x plus fifteen here's another problem that you could try nine x plus five minus three x plus eight go ahead and simplify the algebraic expression nine x minus three x is equal to six x and five plus eight well that's thirteen and so this is the answer six x plus thirteen now let's move on to solving simple linear equations so here's an example x plus 4 is equal to 11. what is the value of x so x is simply a number which you currently don't know the value of so ask yourself what number plus 4 is equal to 11. intuitively you know that 7 plus 4 is 11. so therefore x has to be equal to seven but what can you do to show that x is equal to seven you understand that seven plus four is eleven but mathematically how do you show that in order to find the value of x you need to isolate x you need to get it by itself on one side of the equation and all other numbers you must move to the other side of the equation so we need to get rid of this 4 on the left side the opposite of addition is subtraction so if we subtract both sides by 4 we can get rid of the positive 4 on the left 4 plus negative 4 is 0 and eleven minus four is seven any number added to zero will be equal to that number so x plus zero is simply x therefore x is equal to seven here's one you should work on y plus 5 is equal to negative 4. what is the value of y well just like before we need to isolate y we need to get the y variable by itself and so to remove the positive 5 on the left we need to subtract both sides by 5. so positive 5 and negative 5 adds up to 0 which is nothing so what we have left over on the left side is simply y on the right side we have negative 4 plus negative 5 or simply negative 4 minus five which is equal to negative nine if you use the number line technique if you start with negative four and travel five units to the left you should get negative nine this is negative 5 negative 6 negative 7 negative 8 negative 9. let's say that 12 is equal to x minus 8. what is the value of x so x doesn't have to be on the left side it can be on the right side by itself if we want to find the value of it so we got to move the negative 8 we need to get rid of it on the right side so the opposite of subtraction is addition so let's add 8 to both sides so this will cancel we could bring down the x and on the left side we have 12 plus 8 which is 20. and so that is the value of x now what about this one 3y is equal to 18 what is the value of y so we need to separate 3 from y currently the 3 is multiplied to 1. the opposite of multiplication is division so therefore we need to divide both sides by 3. 3 divided by 3 is 1 and 18 divided by 3 is 6. 1y is the same as y so therefore y is equal to six so if you look at this expression three times what number is eighteen we know that three times six is eighteen so therefore y is equivalent to six now what if you saw an example like this eight is equal to x divided by four what should you do to find the value of x so x is divided by four and the opposite of division is multiplication therefore we need to multiply both sides by four and that's how we can get rid of the four on the right side 4 divided by 4 is 1 and so we just have x on the right side on the left we have 4 times 8 which is 32 so x is 32. now what about that one two thirds x is equal to nine how can we find the value of x if you have a fraction in front of the variable that you want to isolate multiply both sides by the reciprocal of the fraction so that is multiply both sides by three over two nine is the same as nine over one now whatever you do to the left side you must always do to the right side three divided by three is one and two divided by two is one so the twos and threes cancel on the left on the right we have 9 times 3 which is 27 and 1 times 2 which is 2. so the answer is 27 divided by 2. now we could simplify this fraction if we want to 27 is 26 plus 1 and 26 divided by 2 is 13 13 plus a half is 13 and one half as a mixed number so as a decimal this is equal to 13.5 so that is the value of x in this problem try this one x plus 3 divided by 4 is equal to 10 over 5. now what can we do to find the value of x if we have two fractions separated by an equal sign if you see this the best thing you could do is cross multiply four times ten is forty and five times x plus three we need to distribute the five so five times x is five x and five times three is fifteen so this is what we now have our next step is to subtract both sides by 15. 40 minus 15 is 25 so 25 is equal to 5x next we need to divide both sides by 5. 25 divided by 5 is 5 so x is equal to that number go ahead and try this in that last example we solved an equation that looks like this after cross multiplying so this is a multi-step equation now before separating 8 and x you need to get rid of the 5 on the left side so the opposite of addition is subtraction 21 minus 5 is equal to 16 and now we need to divide both sides by 8 to separate x from 8. so 16 divided by 8 is 2 and that is the value of x in this example and we can check it 8 times 2 plus five is that equal to twenty one eight times two is sixteen sixteen plus five is twenty one so x is indeed equal to two go ahead and try this one let's say that we have three plus x divided by four and let's say that's equal to five what is the value of x there's many ways in which you could solve it but if you want to get rid of the fraction multiply everything by four so four times three is twelve x divided by four times four the fours will cancel leaving behind x and then we have four times five which is twenty and now all we need to do is subtract both sides by twelve twenty minus 12 is 8 and so that is the value of x so if you're solving a linear equation and if you have fractions it's helpful to multiply every term by the denominator of the fraction just to clear away all fractions now what about this one sometimes you may have multiple fractions what is the value of x in this case multiply by a multiple of two three and four twelve is the least common multiple of two three and four twelve is divisible by two three and four so first let's multiply twelve by x divided by three so that's going to be 12x divided by three which is four x and then let's multiply 12 by one half half of 12 is six now what is 12 times five fourths there's two ways in which you can do this you can multiply first and then divide or divide first and then multiply twelve times five is sixteen sixty divided by four is fifteen or you could say 12 divided by 4 is 3 3 times 5 is 15. so either case you're going to get the same value now let's subtract both sides by 6. fifteen minus six is nine so we have four x is equal to nine let's divide by four so now we have an improper fraction x is nine over four and that is the answer if you want to convert it to a mixed number separate nine into eight and one a plus one is nine eight divided by four is two so we have two plus one fourth which is the same as two and one fourth as a mixed number as a decimal one fourth is point two five so nine over four is equivalent to 2.25 now let's spend a moment talking about exponents so what is 2 raised to the third power what is that equal to having exponents suggests repeated multiplication and if you recall multiplication is repeated addition so 2 to the third power means that you're multiplying three twos together which is equal to eight four to the third means that you're multiplying four times four times four four times four is sixteen sixteen times four is sixty-four so 4 to the third is 64. now what is the value of these three expressions negative 2 squared negative 3 squared and negative 3 inside of parentheses squared so above we have a negative and we're multiplying two twos the two is positive and we have two of them so 2 times 2 is 4 combined with a negative sign that's negative 4. these two expressions are equivalent so this is negative times 3 times another 3 which is negative 9. on the bottom we have two negative threes multiplied to each other since the negative is inside the parentheses it's affected by the exponent negative three times negative three is positive nine so make sure you know the difference between those expressions now the next thing we need to talk about is factoring binomials for example let's say if we have the expression 14x how can we factor this monomial that is writing everything in terms of prime numbers 14 we can break it down into seven times two and we only have one x variable so that's 14x that's how you factor it now let's say if we want to factor nine y squared nine is three times three y squared is y times y so that's how you could factor that monomial now what about 8x y squared go ahead and factor it completely eight is basically two times two times two we have one x variable and two y variables so that's 8 x y squared completely factored try these two 28 a squared b negative 12 x cubed y and 18 x to the fourth y to the fifth so go ahead and factor those binomials completely let's start with 28 28 is 7 times 4 and 4 we can break that down into 2 times 2 so that's 28 a squared is 8 times a and then we have one b variable 12 is negative four times three and four is two times two x cubed is x times x times x and we have a y variable now eighteen is three times six and six we could break them into three times two x to the fourth means that we're multiplying four x variables and y to the fifth means that we're multiplying five y variables together and so that is the answer so now you know how to factor binomials completely now the next topic of discussion is finding the gcf the greatest common factor what is the greatest common factor between 8 and 12 so looking for a number that's less than 8 and 12 and that goes into 8 and 12. so this number 8 and 12 are both divisible by this integer so what is the highest number that is divisible by 8 and 12. so first let's factor 8 completely eight is two times two times two twelve is two times two times three so notice that eight and twelve have these numbers in common that is two times two so basically it's four that is the greatest common factor between a and twelve eight is divisible by four and twelve is also divisible by 4. let's try another example what is the greatest common factor between 12 and 18 so feel free to pause the video and try that example so let's find out the prime factorization of 12 and 18. 12 is 3 times 4 and 4 is 2 times 2 18 is 3 times 6 and 6 is 3 times two so 12 and 18 have a three in common and they also have a two in common three times two is six so that is the gcf between 12 and 18. the greatest common factor is six now what is the greatest common factor between three numbers 27 36 and 45 so go ahead and try that 27 is 9 times 3 and 9 is 3 times 3 so 27 is 3 to the third power 36 is 3 times twelve and twelve is three times four and four is two times two forty-five is five times nine and nine is three times three so all of these numbers have these two in common that is three times three so the gcf between 27 36 and 45 is nine each of those numbers are divisible by 9. now what about this example what is the greatest common factor between 5 x y and 10 x squared y so let's follow the same process five x y simply five times x times y ten x squared is five times two times x times x times y so we have a five in common and we have an x in common and we also have a y in common so therefore the greatest common factor is five x y let's try this one six x and nine x squared what's the gcf so we could factor six x into three times two times x nine x squared is three times three times x times x so these two terms have 3x in common so that's going to be the gcf between 6x and 9x squared it's 3x now let's spend a few moments simplifying fractions for example what is 14 x squared y divided by 63 xy when dividing monomials what you can do is you can simplify it by factoring 14 is 7 times 2 x squared is x times x and then we have a y 63 is 7 times 9 and we still have an x and a y notice what we can cancel at this point we could cancel a 7 and we can cancel an x and a y so on top what we have left over is 2 times x on the bottom simply a 9. so the answer is 2x divided by 9. and that's how you could simplify monomials by factoring let's try another example what is x squared divided by x to the fifth power so let's simplify by factoring x squared is x times x x to the fifth is basically five x variables multiplied to each other so we could cancel two of them and that leaves behind three x variables on the bottom and x times x times x is simply x cubed so the answer is one divided by x cubed now what about this one y to the fourth divided by y squared y to the fourth is y times y times y times another one and y squared is simply y times y so we can cancel two of these leaving behind y times y which is y squared try this one 21x y squared divided by 28 x squared y cubed so feel free to take a moment to simplify that expression 21 is 7 times 3 and y squared is y times y 28 is 7 times 4 times x squared and y cube is y times y times y so first we can cancel a 7 and we can cancel an x variable and we could cancel two y variables so on top all we have left over is a three and on the bottom we have a four an x and one y variable so it's going to be 4xy and that's the answer so now you know how to simplify uh monomials when they're divided against each other now i'm going to show you an online algebra course that you can use to help you with other topics so if you go to udemy.com and just type in algebra and the course that i created will come up and it's basically this one with a black background so if you go to it you can see an overview and if you go to course content you can see a list of topics that are in this course so i have basic arithmetic addition subtraction multiplication things like that if you want to review your fractions you can look at section three solve linear equations you have a multiple choice quiz as well order of operations graphing linear equations qualities absolute value expressions and there's more polynomials multiply and dividing things like that a whole section on factoring that's a big thing in algebra and then you have systems of linear equations solving by elimination substitution even graphing those things and then you have quadratic equations rational expressions radical expressions and then complex imaginary numbers exponential functions logs how to simplify them functions in general like inverse functions composite functions and then conic sections graphene circles ellipse parabolas hyperbolas there's two video quizzes on that and finally arithmetic and geometric sequences so if you need help in any of these topics feel free to check out this course when you get a chance now let's talk about multiplying monomials what is x squared times x cubed what is that equal to when multiplying monomials you need to add the exponents two plus three is five so it's x to the fifth you can see it this way x squared is x times x x cubed is x times x times x so notice that we're multiplying five x variables together so it's x to the fifth so try these x to the fourth times x to the seventh and x to the 8th times x to the 12th go ahead and try those two problems so this is going to be 4 plus 7 which is 11. and x to the 8th times x to the 12 that's going to be x to the 8 plus 12 which is x raised to the 20th power so when multiplying binomials you should add similar variables exponents here's another example let's say if we have x cube y to the fifth multiplied by x to the sixth y to the eighth so first we need to multiply x cube and x to the sixth and 3 plus 6 is 9 so that's going to be x to the ninth power and here we have y to the fifth times y to the eighth so that's going to be y to the 13th power so we have to add all the exponents now what about this one three x squared times negative four x to the fourth power go ahead and try that so first we gotta multiply three and negative four which that's to be negative 12. and then we can multiply x squared by x to the fourth which is x to the sixth power so it's negative 12 times x to the sixth here's another one two x cubed y to the fourth times eight x to the fifth y to the seventh go ahead and multiply those two terms so let's begin by multiplying two times eight two times a is sixteen next x to the third times x to the fifth three plus five is eight and then y to the fourth times y to the seventh four plus seven is eleven and so you should get that answer now let's talk about dividing monomials what is y to the seventh divided by y squared when multiplying you should add the exponents but when dividing you need to subtract the exponents so this is going to be 7 minus 2 which is 5. now to explain it let's use factorization y to the seventh means that we have seven y variables multiplied together y squared is just y times y we could cancel two of them but notice that we have five y variables left over on the top so that's why it's simply y to the fifth over one which is y to the fifth now what is three to the seventh divided by three to the third power go ahead and try that so we know that we need to subtract the exponents seven minus three is four so this is three to the fourth power which is three times three times three times three three times three is nine so we have nine times nine which is eighty one and so that's the answer for this example now what is x cubed times x to the eighth divided by x to the fifth power so to begin in order to simplify this expression let's multiply x cubed by x to the eighth first three plus eight is eleven now at this point we could divide eleven minus five is six and so that's going to be the final answer here's another one that you could try y to the eighth divided by y squared times y cubed so take a minute and work on that example so first i would multiply the two on the bottom which i think is easiest to do first two plus three is five and now it's best to divide eight minus five is three so the answer is y to the third now try this one what is x squared divided by x to the seventh that's going to be two minus seven you take the top number first and subtract it by the one on the bottom now this is equal to x to the negative five so what does that mean x to the negative five is one over x to the five so when you have a negative exponent we need to do is move the variable to the bottom and the negative exponent will change sign it's going to become positive to verify we could simplify this another way x squared is x times x and x to the seven who knows basically seven x variables multiplied to each other two of which can be cancelled so we have five x variables on the bottom thus is one over x to the fifth power so what is 3 to the negative 1 power so right now the 3 is on the numerator of the fraction if you bring it to the denominator it's going to have a positive 1 exponent x to the negative 2 is equivalent to one over x squared and one over x to the negative four is x to the positive four so when you move a variable or number from the top to the bottom or to the bottom to the top and the exponent changes sign it can switch from negative to positive what is negative 4 raised to the negative 2 power so first we need to bring it down this is negative 4 raised to the second power negative 4 squared is positive 16 so that's one over 16. so now you know what to do if you ever have a negative exponent now let's spend a few minutes talking about percentages so what is 15 of 300 how do you find a percentage of a number well let's see if we could do it mentally so first what is ten percent of three hundred do you know it's very easy to find ten percent of a number all you need to do is move the decimal one unit to the left it's basically one tenth of that number so one half of 300 is 30. so if 10 of 300 is 30 what is five percent of three hundred well five percent is half of ten percent so half of thirty is fifteen so now what is fifteen percent fifteen percent is the sum of ten percent and five percent so therefore fifteen percent is going to be thirty plus fifteen or forty five so forty five is fifteen percent of three hundred now if you want to use your calculator all you need to do is take 300 and multiply by the decimal value of 15 to convert a percentage into a number you can divide this number by 100 or simply move the decimal two units to the left so fifteen percent is equivalent to point one five and so if you take three hundred and multiply it by point one five this will give you forty 45. so that's how you could find the percentage of a number let's try another example what is 20 of 500 see if you could do it mentally now let's find out the value of 10 percent of 500 so all i need to do is move the decimal one unit to the left so 10 percent of 500 is just 50. now 20 of 500 has to be what number well 20 is twice the value of 10 so if we multiply 50 by 2 we'll get 100 so a hundred is twenty percent fifty to verify it multiply five hundred by point twenty and you should get one hundred now here's another one what is 25 percent of 400 so go ahead and try that one mentally so let's find the value of 10 10 of 400 is 40. so another 10 is 40 as well and 5 that's half of 10 so half of 40 is 20. so if we add 10 10 and 5 that will give us the 25 percent that we need and 40 plus 40 plus 20 is a hundred so therefore 25 of 400 which is basically a quarter of 400 or one fourth of it is a hundred here's another example what is 23 percent of 800 so first let's find the value of ten percent ten percent of eight hundred is eighty another ten percent is eighty as well now what is one percent of eight hundred to find one percent you need to move the decimal two units to the left so that's gonna be eight it's basically one tenth of eighty so if one percent is eight what's three percent that has to be eight times three it's three times this value so it's 24. so therefore 23 is the sum of 10 10 and 3. so we get to add up 80 plus 80 which is 160 plus 24 so that's 184 so that's 23 of 800 it's 184. now what is 17 of 900 go ahead and figure that out so let's start with 10 10 of 900 is ninety five percent is half of ten percent so half of ninety is forty five and one percent that's one tenth of ninety so that's gonna be nine so two percent must be twice the value of nine so that's eighteen so to get seventeen percent we gotta add ten five and two that's seventeen percent so we need to add up 90 plus 45 that's 135 plus 18 which is 153 so that should be the answer and to check it you can type this in your calculator take 900 and multiply by 0.17 and you do indeed get 153 now before we end this video there's one more topic that is common in pre-algebra and that's solving similar triangles so let's say if these two triangles are similar to each other and let's say this is 15 this is 12 and this is 9 and this is x if these are similar triangles what is the value of x now the best way to solve a similar triangle is to set up a proportion let's call this triangle one triangle two and let's say that 15 is the height of triangle one 12 is the base and here this is the base and this is the height so let's set up a proportion between triangle one and triangle two so we need two fractions separated by an equal sign on top i'm going to put the height on the bottom the base so the height of triangle one is 15. the height of triangle two is nine the base of triangle one is twelve and the base of triangle two is x and then you're simply solving you have to be careful when setting up the proportion uh correctly if you don't do it correctly then you're gonna get the wrong answer so that's why it helps to have one side to represent triangle one and the other side to represent triangle two now let's cross multiply so we have 12 times 9 which is equal to 15 times x so what is the value of x well let's simplify the before we multiply 12 is 4 times 3 and 9 is 3 times 3. 15 is five times three so we could at least cancel a three now let's divide both sides by five so on the right side we have x on the left side we have four times nine which is three times three divided by five so it's going to be 36 divided by five so that's the value of x now let's turn this into a mixed number 36 is 35 plus one and 35 divided by five is seven so as the mixed number this is seven and one-fifth one-fifth is point two as a decimal so it's also equal to 7.2 you