in this video we're going to focus on pre-alggebra 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 + 3 now many of you know that 5 + 3 is 8 but if you ever have difficulty with this type of math use a 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 1 2 3 this is 6 7 8 so therefore 5 + 3 is 8 now let's try some more examples what is -4 + 5 let's use the number line to get the answer so we're going to start with4 and we're going to add 52 in so we're going to travel 5 units to the right 1 2 3 4 5 this is -3 -21 0 1 so4 + 5 is pos 1 let's work on another example 7 - 5 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 1 2 3 4 5 so this is 6 5 4 3 2 so 7 - 5 is 2 now what about -4 - 2 so let's start with4 and we're going to subtract it by two so we need to go two units to the left and actually that should be 5.3 is on the right side so this is -6 therefore -4 - 2 is equal to -6 now what about this one -6 - -3 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 -6 + 3 so if we're adding we need to travel to the right 1 2 3 this is543 and so -6 + 3 is -3 now what about 8 + -5 what's the answer for this one so if we start at 8 and we're subtracting it by five by the way this expression is equal to 8 - 5 a positive time a negative is a negative sign so we need to travel five units to the left this is going to be 7 6 5 4 3 so 8 - 5 is pos3 now let's talk about multiplication what is 8 * 3 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 8 time 3 means that you're adding 8 three times it's also equivalent to adding 3 eight times but it's easier to add 8 three times 8 + 8 is 16 and 16 + 8 is 24 so therefore 8 * 3 is 24 let's work on another example what's 9 * 4 9 * 4 is equivalent to adding 9 four times 9 + 9 is 18 so these two 9ines add up to 18 and the other two 9ines add up to 18 as well and 18 + 18 is 36 so therefore 9 * 4 is 36 now what is a -5 * 3 a negative * 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 5 + 5 is 10 and 10 + 5 is 15 so therefore,5 * 3 is5 try this one what is -6 multiplied by8 when you multiply two negative numbers you're going to get a positive result so this is equivalent to multiplying 6 * 8 so I'm going to add 8 six times instead of adding six eight times now adding 2 eights will give me 16 so I have 16 + 16 + 16 and 16 + 16 is 32 and 32 + 16 is 48 so therefore -6 *8 is equal to this number pos 48 now let's move on to our next example let's focus on division what is 54 / 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 * 9 is 54 so 54 / 6 is 9 so division is simply the opposite of multiplication so here's another example what is - 455 / pos 9 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 9 so 9 * what number is equal to 45 it turns out that you have to add 9 five times to get to 45 9 + 9 is 18 18 + 9 is 27 27 + 9 is 36 36 + 9 is 45 so therefore 9 * 5 is - 455 and if we focus on the reverse statement -45 / 9 that's going to be5 and so that's a quick and simple way to perform simple division here's another example what's -12 / -2 when you divide two negative numbers you're going to get a positive result so this is equivalent to dividing 12 by two so 2 * what number is 12 you have to add 2 six times to get to 12 2 + 2 is 4 if you add another two that's six and then 8 and then 10 and then 12 so therefore 2 * 6 is equal to 12 and 12 / 2 has to be six now let's say if you have this problem what is 8 - 5 * 4 so what is the answer now there's two possible ways of attempting to do this problem and one of the two ways that 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 - 5 is 3 and 3 * 4 is 12 we're going to get that result but now let's say if we multiply first -5 * 4 is -20 so this becomes 8 - 20 and 8 - 20 is -12 so the results are different so which one comes first subtraction or multiplication perhaps you heard of PEMDAUS please excuse my dear aunt Sally p stands for parenthesis 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 pendas to know which operation should come first so therefore this is the correct answer 8 - 5 * 4 is -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 -12 now let's move on to another example try this one what is 6 + 24 / 4 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 / 4 24 / 4 is 6 because 4 * 6 is 24 and 6 + 6 is 12 so that's the final answer in this example now let's try another one what is 8 - 5 * 7 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 parentheses so you're comparing parenthesis to multiplication and you need to work inside the parentheses before you multiply so 8 - 5 is 3 and 3 * 7 is 21 so that's the answer in this particular example now what about this problem what is 24 / 4 * 3 should we perform division first or multiplication now according to the word pendas 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 / 4 is 6 6 * 3 is 18 now let's do it the other way let's perform multiplication first 4 * 3 is 12 and 24 / 12 is 2 so as we could see um 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 PEMDAUS parenthesis has more priority than multiplication and division so in this case we need to work inside the parenthesis 4 * 3 is 12 and so we have 24 / 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 / 8 - 2 * 3 so first we need to work inside the parenthesis 8 - 2 is 6 so we have 48 / 6 * 3 now that we have division and multiplication we need to work starting from the left towards the right 48 / 6 is 8 and 8 * 3 is 24 and so that's going to be the final answer for this problem here's another example what is 32 - 24 / 8 / 2 so feel free to pause the video and simplify this expression 8 / 2 is 4 and 32 - 24 is 8 and 8 / 4 is 2 so that's going to be the final answer in this example try these two problems what's 7 * 9 - 4 and what is 3 * 4 + 8 - 2 / 2 so the one above is simple we need to work inside the parentheses first 9 - 4 is 5 and 7 * 5 is 35 now let's work on this example so first we need to subtract 8 by 2 8 - 2 is 6 and now we need to work inside the brackets 6 / 2 that's equal to 3 so we have 3 4 + 3 now what's our next step 4 + 3 is 7 and 3 * 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 / 2 + 5 and let's say that you're told x is equal to 4 and y is equal to 3 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 4 and y we're going to replace it with three so what we now have is 4 * 3 / 2 + 5 4 * 3 is 12 and 12 / 2 is 6 6 + 5 is 11 so that is the value of this expression given x = 4 and y = 3 let's work on another example evaluate the expression let's say the expression is 4x + 3 y - 2 z 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 five and we're going to replace y with 2 and z with 3 and then just perform the operation 4 * 5 is 20 3 * 2 is 6 2 * 3 is also 6 6 - 6 is 0 and 20 + 0 is simply 20 so therefore that's the value of this expression let's try another example what is 5x - 2 * y + 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 7 and z is 4 so don't forget to perform order of operations we need to add 7 + 4 we could multiply 5 * 3 simultaneously that's going to be 15 7 * 4 is 11 now we need to multiply before we subtract 2 * 11 is 22 so what we have is 15 - 22 which will give you -7 and so that's the end result for this problem try this one x^2 - Y^ 2 / 4 Z + 8 and let's say that x is equal to 8 y is 6 and let's say z is 4 so x^2 will be replaced with 8^2 and y let's replace it with six and then let's substitute z with 4 so this is the expression that we need to simplify so now what is 8^2 8^2 is 8 * 8 which is 64 6^2 or 6 * 6 that's equal to 36 and on the bottom we have 4 * 4 which is 16 now what is 64 - 36 if we use a calculator that's equal to 28 and 16 + 8 is 24 now this fraction is reducible so how can we reduce this improper fraction 28 is 7 * 4 24 is 6 * 4 4 / 4 is 1 so we can cancel it so what we have left over is 7 / 6 and that is the answer what would you do if you saw an expression that looks like this what is 3 * x + 4 so we can't really add x + 4 x is a variable in which uh 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 3 to x and 4 3 * x is simply 3x and 3 * pos4 it's 12 so this expression is equal to 3x + 12 let's try another example now what is 4 * 2x - 3 go ahead and use the distributive property 4 * 2x is equal to 8x and 4 * -3 is -12 so this is equal to 8x - 12 now sometimes you may have some other algebraic expressions to simplify here's another one what is 5x + 3x go ahead and simplify all you need to do is add the coefficients 5 + 3 is 8 so this is equal to 8x now what about this what's 7 y + 2 y + 8 so based on the last example go ahead and simplify this expression what we need to do is add like terms 7 y + 2 y and that's equal to 9 y now we cannot add 9 y and 8 because what is it going to be 17 or 17 y because the 8 doesn't have a y it's not a similar term to 9 y so we cannot add them therefore the final answer is 9 y + 8 try this one 3 * x + 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 3 + 8 is 11 so 3x + 8 x is 11 x therefore the final answer is 11 x + 15 here's another problem that you could try 9x + 5 - 3x + 8 go ahead and simplify the algebraic expression 9x - 3x is equal to 6x and 5 + 8 well that's 13 and so this is the answer 6x + 13 now let's move on to solving simple linear equations so here's an example x + 4 is= 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 + 4 is equal to 11 intuitively you know that 7 + 4 is 11 so therefore x has to be equal to 7 but what can you do to show that x is equal to 7 you understand that 7 + 4 is 11 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 four on the left side the opposite of addition is subtraction so if we subtract both sides by four we can get rid of the positive four on the left 4 + -4 is 0 and 11 - 4 is 7 any number added to zero will be equal to that number so x + 0 is simply x therefore x is equal to 7 here's one you should work on y + 5 is equal to -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 five on the left we need to subtract both sides by five so pos5 and neg 5 adds up to zero which is nothing so what we have left over on the left side is simply y on the right side we have -4 +5 or simply4 minus 5 which is equal to9 if you use the number line technique if you start with4 and travel 5 units to the left you should get9 this is5 -6 -789 let's say that 12 is equal to x - 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 + 8 which is 20 and so that is the value of x now what about this one 3 y is equal to 18 what is the value of y so we need to separate three from y currently the 3 is multiplied to y the opposite of multiplication is division so therefore we need to divide both sides by 3 3 / 3 is 1 and 18 / 3 is 6 1 y is the same as y so therefore y is equal to 6 so if we look at this expression 3 * what number is 18 we know that 3 * 6 is 18 so therefore y is equivalent to 6 now what if you saw an example like this 8 is equal to x / 4 what should you do to find the value of x so x is divided by 4 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 four divided 4 is one and so we just have x on the right side on the left we have 4 * 8 which is 32 so x is 32 now what about that one 2/3 x is equal to 9 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 3 over2 9 is the same as 9 over 1 now whatever you do to the left side you must always do to the right side 3 / 3 is one and two divided two is one so the twos and threes cancel on the left on the right we have 9 * 3 which is 27 and 1 * 2 which is 2 so the answer is 27 / 2 now we could simplify this fraction if we want to 27 is 26 + 1 and 26 / 2 is 13 13 plus a half is 13 and 12 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 + 3 / 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 4 * 10 is 40 and 5 * x + 3 we need to distribute the 5 so 5 * x is 5 x and 5 * 3 is 15 so this is what we now have our next step is to subtract both sides by 15 40 - 15 is 25 so 25 is equal to 5x next we need to divide both sides by 5 25 / 5 is five 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 five on the left side so the opposite of addition is subtraction 21 - 5 is equal to 16 and now we'll need to divide both sides by 8 to separate x from 8 so 16 / 8 is 2 and that is the value of x in this example and we can check it 8 * 2 + 5 is that equal to 21 8 * 2 is 16 16 + 5 is 21 so x is indeed equal to 2 go ahead and try this one let's say that we have 3 + x / 4 and let's say that's equal to 5 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 4 so 4 * 3 is 12 x / 4 * 4 the fours will cancel leaving behind x and then we have 4 * 5 which is 20 and now all we need to do is subtract both sides by 12 20 - 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 2 3 and 4 12 is the least common multiple of 2 3 and 4 12 is divisible by 2 3 and 4 so first let's multiply 12 by x / 3 so that's going to be 12x / 3 which is 4x and then let's multiply 12 by 12 half of 12 is 6 now what is 12 * 5/4s there's two ways in which you can do this you can multiply first and then divide or divide first and then multiply 12 * 5 is 60 60 / 4 is 15 or you could say 12 / 4 is 3 3 * 5 is 15 so either case you're going to get the same value now let's subtract both sides by six 15 - 6 is 9 so we have 4x is equal to 9 let's divide by 4 so now we have an improper fraction x is 9 / 4 and that is the answer if you want to convert it to a mixed number separate 9 into 8 and 1 8 + 1 is 9 8 / 4 is 2 so we have 2 + 1/4 which is the same as 2 and 1/4 as a mixed number as a decimal 1/4 is 0.25 so 94 is equivalent to 2.25 now let's spend a moment talking about exponents so what is 2 raised to the 3 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 8 4 the 3r means that you're multiplying 4 * 4 * 4 4 * 4 is 16 16 * 4 is 64 so 4 to the 3r is 64 now what is the value of these three expressions -2^2 -3^2 and -3 inside a 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 * 2 is 4 combined with a negative sign that's4 these two expressions are equivalent so this is * 3 *3 which is9 on the bottom we have two -3's multiplied to each other since the negative is inside the parentheses it's affected by the exponent -3 *3 is positive 9 just make sure you know the difference uh between those expressions now the next thing we need to talk about is factoring monomials 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 7 * 2 and we only have one x variable so that's 14x that's how you can factor it now let's say if we want to factor 9 y^ 2 9 is 3 * 3 y^2 is y * y so that's how you can factor that monomial now what about 8x y^2 go ahead and factor it completely 8 is basically 2 * 2 * 2 we have one x variable and two y variables so that's 8x y^2 completely factored try these two 28 a^2 b -12 x cub y and 18 x 4 y 5th so go ahead and factor those monomials completely let's start with 28 28 is 7 * 4 and 4 we can break that down into 2 * 2 so that's 28 a^ 2 is a * A and then we have one B variable 12 is -4 * 3 and 4 is 2 * 2 x cub is X * X * X and we have a Y variable now 18 is 3 * 6 and 6 we could break into 3 * 2 x to the 4th means that we're multiplying four X variables and Y to the 5th means that we're multiplying five Y variables together and so that is the answer so now you know how to factor monomials 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 we're 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 uh by 8 and 12 so first let's factor 8 completely 8 is 2 * 2 * 2 12 is 2 * 2 * 3 so notice that 8 and 12 have these numbers in common that is 2 * 2 so basically it's four that is the greatest common factor between a and 12 8 is divisible by 4 and 12 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 uh example so let's write out the prime factorization of 12 and 18 12 is 3 * 4 and 4 is 2 * 2 18 is 3 * 6 and 6 is 3 * 2 so 12 and 18 have a three in common and they also have a two in common 3 * 2 is 6 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 * 3 and 9 is 3 * 3 so 27 is 3 to the 3 power 36 is 3 * 12 and 12 is 3 * 4 and 4 is 2 * 2 45 is 5 * 9 and 9 is 3 * 3 so all of these numbers have these two in common that is 3 * 3 so the GCF between 27 36 and 45 is 9 each of those numbers are divisible by 9 now what about this example what is the greatest common factor between 5xy and 10 x^2 y so let's follow the same process 5 x y is simply 5 * x * y 10 x^2 is 5 * 2 * x * x * 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 5x y let's try this one 6 x and 9 x^2 what's the GCF so we could factor 6x into 3 * 2 * x 9 x^2 is 3 * 3 * x * x so these two terms have 3x in common so that's going to be the GCF between 6x and 9x^2 it's 3x now let's spend a few moments simplifying fractions for example what is 14 x^2 y / 63 x y when dividing monomials what you can do is you can simplify it by factoring 14 is 7 * 2 x^2 is x * x and then we have a y 63 is 7 * 9 and we still have an x and a y notice what we can cancel at this point we could cancel a seven and we can cancel an x and a y so on top what we have left over is 2 * x on the bottom simply a 9 so the answer is 2x / 9 and that's how you could simplify monomials by factoring let's try another example what is x^2 / x 5th power so let's simplify by factoring x^2 is x * x x 5th is basically 5 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 * x * x is simply x cub so the answer is 1 / x cub now what about this one y 4 / y^2 y 4th is y * y * y * another y and y^2 is simply y * y so we could cancel two of these leaving behind y * y which is y^2 try this one 21x y^2 / 28 x^2 y cub so feel free to take a moment to simplify that expression 21 is 7 * 3 and y^2 is y * y 28 is 7 * 4 * x^2 and y cub is y * y * y so first we can cancel a 7 and we can cancel an x variable and we can 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 4x y 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 uh to help you with other topics so if you go to udemi.com and just type in algebra and the course that I created will come up and it's basically this one uh 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 uh basic arithmetic addition subtraction multiplication things like that if you want to review a fractions you can look at section three solving linear equations you have a multiple choice quiz as well order of operations graphing linear equations linear equalities absolute value expressions and there's more polomials multiplying 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 consections graphing circles ellipse parabas hyperas 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^2 * x cub what is that equal to when multiplying monomials you need to add the exponents 2 + 3 is 5 so it's x 5th you could see it this way x2 is X * X x cub is X * X * X so notice that we're multiplying five X variables together so it's X 5th so try these x 4th * X 7 and X 8 * X 12th go ahead and try those two problems so this is going to be 4 + 7 which is 11 and x^ 8 * x^ 12 that's going to be x^ 8 + 12 which is x raised to the 20th power so when multiplying monomials you should add similar uh variables uh exponents here's another example let's say if we have x cub y 5th multiplied x^ 6 y 8th so first we need to multiply x cub and x 6 and 3 + 6 is 9 so that's going to be x^ the 9th power and here we have y 5th * y 8th so that's going to be y^ the 13th power so we have to add all the exponents now what about this one 3x^2 * -4x 4th power go ahead and try that so first we got to multiply 3 and -4 which that's going to be -12 and then we can multiply x^2 by x 4th which is x^ 6 power so it's -12 * x 6 here's another one 2x cub y 4th* 8 x 5th y 7th go ahead and multiply those two terms so let's begin by multiplying 2 * 8 2 * 8 is 16 next x 3r * x 5th 3 + 5 is 8 and then y 4th * y 7th 4 + 7 is 11 and so you should get that answer now let's talk about dividing monomials what is y to the 7th / y^2 when multiplying you should add the exponents but when dividing you need to subtract the exponents so this is going to be 7 - 2 which is 5 now to explain it let's use factorization y to the 7th means that we have seven y variables multiplied together y^2 is just y * 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 5th over 1 which is y 5th now what is 3 to the 7th / 3 power go ahead and try that so we know that we need to subtract the exponents 7 - 3 is 4 so this is 3 to the 4th power which is 3 * 3 * 3 * 3 * 3 is 9 so we have 9 * 9 which is 81 and so that's the answer for this example now what is x cub * x 8 / x 5th power so to begin in order to simplify this expression let's multiply x cub x 8 first 3 + 8 is 11 now at this point we could divide 11 - 5 is 6 and so that's going to be the final answer here's another one that you could try y 8 / y^2 * yub 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 2 + 3 is 5 and now it's best to divide 8 - 5 is 3 so the answer is y 3 now try this one what is x^2 / x 7th that's going to be 2 - 7 you take the top number first and subtract it by the one on the bottom now this is equal to x^ 1 / x^ 5 so when you have a negative exponent what you 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^2 is X * X and X to the 7 you know is basically seven X variables multiply to each other two of which can be cancelled so we have five X variables on the bottom thus is 1 / X^ 5th power so what is 3 to the 1 power so right now the three 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 -2 is equivalent to 1x^2 and 1 /x -4 is x to the pos4 so when you move a variable or number from the top to the bottom or to the bottom to the top the exponent changes sign it can switch from negative to positive what is -4 raised to the -2 power so first we need to bring it down this is -4 raised to the second power -4^2 is pos6 so that's 1 / 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 10% of 300 do you know it's very easy to find 10% of a number all you need to do is move the decimal one unit to the left is basically one10enth of that number so one of 300 is 30 so if 10% of 300 is 30 what is 5% of 300 well 5% is half of 10% so half of 30 is 15 so now what is 15% 15% is the sum of 10% and 5% so therefore 15% is going to be 30 + 15 or 45 so 45 is 15% of 300 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 15% is equivalent to 0.15 and so if you take 300 and multiply by 0.15 this will give you 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 can do it mentally now let's find out the value of 10% of 500 so all I need to do is move the decimal one unit to the left so 10% 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 two we'll get 100 so 100 is 20% of 50 to verify multiply 500 by 20 and you should get 100 now here's another one what is 25% 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% that we need and 40 + 40 + 20 is 100 so therefore 25% of 400 which is basically a quarter of 400 or 1/4 of it is 100 here's another example what is 23% of 800 so first let's find the value of 10% 10% of 800 is 80 another 10% is 80 as well now what is 1% of 800 to find 1% you need to move the decimal 2 units to the left so that's going to be 8 it's basically 1/10th of 80 so if 1% is 8 what's 3% that has to be 8 * 3 it's 3 times this value so it's 24 so therefore 23% is the sum of 10 10 and 3 so we got to add up 80 + 80 which is 160 + 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 90 5% is half of 10% so half of 90 is 45 and 1% that's 110th of 90 so that's going to be 9 so 2% must be twice the value of 9 so that's 18 so to get 17% we got to add 10 five and two that's 17% so we need to add up 90 + 45 that's 135 + 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.17 and you do indeed get 153 now before we end this video there's one more topic that is common in pre-alggebra 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 1 triangle 2 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 1 is 15 the height of triangle 2 is 9 the base of triangle 1 is 12 and the base of triangle 2 is x and then you're just simply solving you have to be careful when setting up the proportion uh correctly if you don't do it correctly then you're going to 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 * 9 which is equal to 15 * x so what is the value of x well let's simplify before we multiply 12 is 4 * 3 and 9 is 3 * 3 15 is 5 * 3 so we could at least cancel a three now let's divide both sides by 5 so on the right side we have x on the left side we have 4 * 9 which is 3 * 3 / 5 so it's going to be 36 / 5 so that's the value of x now let's turn this into a mixed number 36 is 35 + 1 and 35 / 5 is 7 so as a mixed number this is 7 and 1/5 1/5 is2 as a decimal so it's also equal to 7.2