Brief Summary
This video provides a comprehensive guide on simplifying radicals with variables and exponents. It covers techniques for breaking down radicals, handling exponents, and rationalizing denominators, including when to use absolute values.
- Simplifying radicals with variables and exponents involves breaking down the terms inside the radical and applying exponent rules.
- When dividing variables within a radical, subtract the exponents.
- Rationalizing the denominator is essential to remove radicals from the denominator.
To simplify radicals with variables and exponents, one approach is to expand the variable with its exponent and extract pairs based on the index number. For example, the square root of x to the fifth power can be written as x multiplied by itself five times. Since it's a square root, pairs of x can be taken out, resulting in x squared times the square root of x. An alternative method involves dividing the exponent by the index; the quotient becomes the exponent outside the radical, and the remainder is the exponent of the variable remaining inside the radical.
To simplify radicals with numbers, break down the number into factors, one of which is a perfect square. For example, the square root of 32 can be broken down into the square root of 16 times 2. Since the square root of 16 is 4, the simplified form is 4 times the square root of 2.
When simplifying more complex radicals, such as the square root of 50x³y¹⁸, simplify each term separately. For the variable terms, divide the exponents by the index. If the index is even and the resulting exponent is odd, use absolute value. Simplify the numerical part by factoring out perfect squares. For instance, the square root of 50 can be simplified to 5 times the square root of 2.
To simplify cube roots, divide the exponents of the variables by 3. The quotient becomes the exponent outside the radical, and the remainder stays inside. For example, the cube root of x⁵y⁹z¹⁴ simplifies to xy³z⁴ times the cube root of x²z². When simplifying the cube root of 16, identify perfect cubes that factor into 16, such as 8. The cube root of 8 is 2, so the simplified form includes 2 outside the radical and the cube root of 2 inside.
When simplifying radicals containing fractions, simplify within the radical first by dividing variables with the same base, subtracting exponents. Then, simplify the numerical part by factoring out perfect squares. Rationalize the denominator by multiplying both the numerator and the denominator by the radical in the denominator. Remember to use absolute values when the index is even and the resulting exponent is odd.
To simplify cube roots with fractions, simplify the fraction inside the radical by dividing both the numbers and variables. Then, take the cube root of any perfect cubes. Rationalize the denominator by multiplying the numerator and denominator by the cube root of the denominator to eliminate the radical in the denominator. Absolute values are not needed for odd indices.
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Transcript
Full transcript of “Simplifying Radicals With Variables, Exponents, Fractions, Cube Roots - Algebra”. Click a timestamp to open the video at that moment. Need another video? Use the YouTube transcript generator.
0:00 in this video we're going to focus on how to simplify radicals with variables and exponents so let's say if you want to simplify the square root of x the 5ifth the index number is a two now one way you can do this is you can write x five times and because there a two you need to take out two at a time so this will come out as 1 x and this will come out as another X and you're going to get x * X sare < TK of one just X by itself so this is equal to x^2 otk x now another way you can simplify this or get that same answer is if you do it this way how many times does two go into five two goes into five two times because 2 * 2 is four 2 * 3 is six that's too much and what's remaining 5 minus 4 is 1
0:58 so you get one remaining that's another way you can simplify so let's say for example if you want to simplify the square root of x the 7 how many times does two go into seven two goes into seven three times with one remaining now let's try this one how many times does two go into eight two goes into eight or 8 divided by two is four two goes into eight four times with no remainder two goes into nine four times and two goes into 12 six times with no remainder for the nine there's a remainder of one so the Y is still on the inside that's a quicker way that you can use to simplify radicals now let's say if you have a number let's say if you want to simplify the square root of let's say um 32 what you want to do is break this down
2:01 into um two numbers one of which was is a perfect square so 32 can be broken down into 16 and two now the reason why I chose 16 and two is because we know what the square root of 16 is and that's four and so this is just 4 < tk2 now let's say if we have a problem that looks like this let's say if we want to simplify the square root of 50 x Cub y the 18th so how many times does two go into three two goes into three one time with uh one remaining and two goes into 18 n times now usually when you have an even index and an odd exponent you got to put it in absolute value now your teacher may not go over this but some teachers do but just in case if you have one of those teachers who wants you to use an absolute value you only need it if you have an even index and if you get an odd exponent after it comes out of the
3:10 radical now the only thing we have to simplify is root 50 square otk 50 we can break it down into square < TK 25 and two because 25 * 2 is 50 and the square root of 25 is 5 but the two stays inside the radical so we can put a five on the outside and let's put the two inside so this is the final answer that's how you can simplify that expression let's try some other problems so let's say if we have the cube root of x to the 5th y to the 9th and Z to the 14th so how many times does three go into five three goes into five one time with uh two remaining so we're going to put X squ inside and the index number would it's going to stay three now how many times is three going to 9 9 ID 3 is three with no remainder and how many times does three go into 14 three goes into 14 four times and 3 * 4 is 12 so 14 - 12 is two so we have two remaining so that's how you can simplify radicals let's try one final problem thr
4:29 pause the video and see if you see if you can get the answer for this one so the cube root of 16 x to the 14th y to the 15th Z to the 20th so how many times is three go to 14 three goes into 14 four times 3 * 4 is 12 and 14 - 12 is 2 so we're going to get x s on the inside 3 goes into 15 five times with no remainder because 15 / 3 is 5 3 goes into 206 times 3 * 6 is 18 three doesn't go into 20 evenly and 20 - 18 is 2 so there's two remaining now let's simplify the cube root of
5:19 16 perfect cubes are one one cube is one 2 to the 3 power is 8 3 to the 3 power is 27 so a perfect Cube that goes into 16 is 8 so 16 / 8 is 2 so you want to write cube root of 16 as the cube root of8 time cubot of two because the cube root of 8 simplifies the two so this two is going to go on the outside which we're going to put it here and this two remains on the inside which I'm going to put it there so this is our final answer for that problem so that's how you can simplify radicals with variables in exponents but actually let's try one more let's say if you have a question that looks like this let's say the square root of 75 x to the 7th y the 3 Z to the 10th over 8 let's say x to the 3 y to the 9th Z to the 4th so the first thing we can do is um let's simplify everything let's rewrite it so 75 is uh 25 time 3 we can square root 25 that's five but we'll do that later and 8 is 4 * 2 because we can take the square root of four now when you divide exponents I
6:59 mean when you divide variables you got to subtract the exponents 7 - 3 is 4 and that goes on top because there's more X values on top now for this one you can do 3 - 9 but I think it's easier if you subtract it backwards the 9 minus 3 which is six and because we subtract it backwards the six goes on the bottom and then 10 minus 4 so there's more Z's on top than on the bottom so we're going to put it on top so Z to the 6 and now let's simplify the square OT of 25 is five and the square root of 4 is two now two goes into four two times 4id two is two so we get x2 2 goes into six three times so we get Z to the 3 and 6 ID 2 is three so we get y the 3 and inside the radical we still have a radical 3 and a < tk2 left over so now we also need to add some absolute values because we have an e even index and we have a few OD exponents we need to put Z in absolute value and a y so our last step is to multiply top and bottom by square root of two we need to
8:16 rationalize the denominator we need to get rid of that radical so our answer our final answer is 5 x s absolute value of Z to the 3qu < TK 6 over < tk2 * < tk2 is s < TK 4 which simplifies to two and 2 * 2 gives us four so we get four absolute value y Cub this is our final answer for that particular problem okay let's try just one more problem so let's say if we have um the cube root of 16 x to the 7 y 4th is Z to the 9th / 54 x^2 y to the 9th Z to the 15th so feel free to pause the video and
9:15 try this example yourself so the first thing I would do is within a radical I would divide both numbers by two referring to the 16 and the 54 so right now what I have is the cube root of 8 which is a perfect Cube over 27 16 ID 2 is 8 half of 54 is 27 so now what I'm going to do is subtract the exponents 7 - 2 is five and for the Y's I'm going to subtract it backwards 9 - 4 is five so y to the 5th and for Z I'm going to subtract it backwards 15 minus 9 is six but that's going to go on the
10:05 bottom and so now we could simplify so the cube root of 8 is two and three goes into five only one time with two remaining and the cube roet of 27 is three and 3 goes into 5 1 * time just like x with two remaining three goes into six two times so that becomes z^2 now we don't need any absolute values because this is an odd index we only need it for even index numbers that produce an odd exponent so now let's simplify what we have so we need to get rid of the radical on
10:55 the bottom so we're going to multiply top and bottom by the cube root of y to the first power so what we now have is 2x cuot x^2 * y / 3 y * Z ^ 2 times the cube root of y 3 the cube root of Y the 3 cancels and so that becomes y the 1 and y 1 * y 1st is y^ 2 so our final answer is 2X cube root x^2 y over 3 y^2 z^2 and that's it so now you know how to simplify radicals with variables and exponents so that's it for this video thanks for watching and uh have a have a wonderful day
