in this video we're going to talk about how to simplify fractional exponents so let's go ahead and begin let's say if we have 8 raised to the 2 over 3 power what is that equal to how can we find the value of this expression if you know the answer or if you know how to do it feel free to pause the video and try now the best way to answer it is to separate the fraction into two parts so this equivalent to 8 raised to the 1/3 power which is raised to the second power 1/3 * 2 is 2/3 so what is 8 raised to the 1/3 whenever you raise something to the 1/3 power you're basically looking for the cube root so what is the cube root of 8 what number times itself three times is8 so what time what time what is 8 it turns out that the cube root of 8 is 2 2 * 2 * 2 3 * is eight and so now we got to figure out what 2^ squ is 2^2 is basically taking two twos and multiplying them together 2^ 2 is equal to 4 and so 8 raised to the 2/3 power is 4 so now that you know how to do that one go ahead and try this one calculate the value of 16 raised to the 54s so first let's find a four root of 16 and then let's raise it to the fifth power so what number times itself four * is equal to 16 the 4ot of 16 is 2 because 2 * 2 * 2 * 2 4 * is 16 now what is the value of 2 to the 5th power 2 * 2 * 2 * 2 * 2 5 * is equal to 32 and so that's the answer let's try another example calculate the value of 9 raised to 3 over2 so once again we're going to find a square root of 9 which is 9^ the 12 and then we're going to raise it to the thir power the square root of 9 is three and 3 to the 3 power which is basically 3 * 3 * 3 3 * 3 * 3 is 9 9 * 3 is 27 so 9^ the 3es is 27 what is the cube root of 64 s what can we do if we have a radical and an exponent on the inside well you can change it into its exponential form you can write it as a fractional exponent this is equivalent to 64 raised to the 2/3 which is the cube root of 64 raised to the second power now what is the cube root of 64 what number times itself three times is 64 four * 4 * 4 that's 64 so we're going to have a 4^ 2ar and 4 2 or 4 * 4 is 16 so the cube root of 64 s is simply 16 try this one calculate the four root of 81 Cub so let's rewrite it this is 81 raised to the 34s which we can find a four root first and then we'll raise it to the third power the fourth root of 81 is three because 3 * 3 * 3 * 3 four times is 81 and 3 to the 3 we've uh calculated this value already it's 27 here's another one that we could try what about the fifth root of 32 raised to the 4th power so just like before we can rewrite it like this so we need to find the fifth root first and then raise it to the fourth power the fifth root of 32 is five I mean not five but two and 2 to the 4th power that's two 2 * 2 * 2 * 2 2 * 2 is 4 and those twos will also make a four 4 * 4 is 16 so 16 is the final answer now what would you do if we have a negative exponent let's say 25 raised to the 3 over2 what is that equal if you have a negative exponent you can make it positive by taking a 25 and putting it to the Bottom now first we need to find the square root of 25 and then we need to raise it to the third power the square OT of 25 is 5 and 5 raised to the 3 power 5 * 5 * 5 3 * that's 125 so the final answer is 1 / 125 and so that's it now how would you simplify this expression what is the cube root of x to the 4th power multiplied by the four root of x to the 5th power what would you do here now we can't multiply x 4 * x 5 the reason being is the index numbers are not the same if they were the same we could do that but since they're not the same we can't so how can we simplify this expression in order to do so we need to convert the radical expression to a fractional exponent so the cube root of x 4th is the same as X raised to 4/3 and the other one is going to be X raised to 54s now let's go over some basic rules of algebra whenever you multiply by Common base you may add the exponents so x^2 * X Cub is X 5th power 2 + 3 is 5 so in this case we need to add the two fractions and to add them you need to get common denominators so let's find out what 4 over 3 + 5 over 4 is equal to the least common multiple of 3 and four is 12 so let's multiply this fraction by 3 over three and the other one by four over 4 so this is going to equal 16 over 12+ 15 over 12 16 + 15 is uh 31 so this is 31 over2 so thus we have X raised to the 31 over2 which we can rewrite it as the 12th root of x^ the 31 power now if we want to we can simplify uh that radical it's not fully simplified but let's go ahead and simplify now how many times does 12 go into 31 12 goes into 31 two times 12 * 2 is 24 but first i'm going to split it into two radicals 24 and 7 adds up to 31 now 24 / 12 is 2 so this is going to be x^2 * the 12th root of x 7 so if you want to simplify it that's the final answer let's try this one what is the uh fifth root of x Cub / let's say the fourth root of x to 7 try this one go ahead and simplify so first let's convert it into a fractional exponent this is going to be X raised to 3 over 5 / X ra to 7 over 4 now when you're dividing by two common bases you need to subtract the exponents 7 - 2 is 5 so in this case we got to subtract the fractions so we have 3 over 5 - 7 over4 so let's go ahead and get common denominators let's multiply this fraction by 5 over five and the other one by 4 over 4 so this will give us a common denominator of 20 4 * 3 is 12 4 * 5 is 20 7 * 5 is 35 and 12 - 35 is -23 over 20 so that's what we have at this point x raised to the -23 over 20 which is 1 over x to the 23rd over 20 which if we convert it back to radical form this is the 20th root of x to the 23rd 20 goes into 23 one time so we can take out an X and 23 minus 20 is three so there's three left over but the index number will remain 20 but we have a three left over on the inside now some teachers may require you to put an absolute value if you have an even index number and if you get an odd exponent a variable coming out of the radical you need to enclose it in parenthesis I mean not parenthesis but absolute value so that you should write it as absolute value of x times the 20th root of x Cub so this is going to be the final answer to this problem and so that's it for this video if you want to find more videos on algebra exponents order of operations feel free to check out my channel look for my playlists and you can find topics on chemistry physics and you can check out my website video- the.net you can also access all of my playlists there as well so thanks for watching and have a great day