Perform the indicated operations. Assume all variables represent ...

Question

Perform the indicated operations. Assume all variables represent positive real numbers. 2∛3 + 4∛24 - ∛81

Accepted Answer

Hello. Today we're going to be simplifying the given expression. So what we are given is five times the cube root of 128 And we're going to be subtracting three times the cube root of 16. Then adding two times the cube root of 54. Now currently 1 28, 16 and 54 are not perfect cubes. But let's go ahead and list out a perfect couple perfect cubes that we can use to help us simplify this expression. The first perfect You we can use is eight as eight can be rewritten as two times itself three times. Which can also be rewritten as too cute. The next perfect cube we can work with is 27 as can be rewritten as three times itself three times which can be rewritten as three cubed. And finally the last one we want to work with is 64 as 64 can be rewritten as four times itself four times which itself can be rewritten as four cubed. So if we wanted to simplify this expression let's go ahead and rewrite the current inner arguments of the Cube roots. We can rewrite 128 to be 64 times two. So what we have in the first cube root is five times the cube root of 64 times two. In the second cube root we can rewrite 16 as eight times two. So what we have in the second term is negative three times the cube root of eight times two. And in the last term 54 can be rewritten as 27 times two. So what we have is two times the cube root of 27 times two. Now notice that we have 64 8 and 27. All three of these are perfect cubes that we stated earlier. So we can replace 64 with four cubed, we can replace eight with two cubed and we can replace 27 with three cubed. So if we do that we can rewrite the expression to be five times the cube root of four cubed times two -3 times the cube root of two cubed times two plus two times the cube root of three cubed times two. Now the cube root allows you to take out any numbers that are in multiples of three. And here we have four times itself three times we have two times itself three times and we have three times itself three times. So we can go ahead and take out 42 and three from the cube roots. And we can rewrite that as five times four times the cube root of two -3 times two times the cube root of two Plus two times Times The Cube Root of two. And now we just need to go ahead and simplify the coefficients five times four is going to give us 20. So the first term can be rewritten as 20 times the cube root of two, three times two is going to give us six. So we have negative six times the Q word of two. And for the last term two times three is going to give us six. So we have positive six times the cube root of two. Now we just need to go ahead and combine the cube roots together since we have the same arguments of the cube root of two in each term, but notice that we have the coefficient of negative six in the second term and the coefficient of positive six in the third term. If we were to add these two terms together, what's going to happen is that these two terms are going to cancel each other out and what we are left with is 20 times the cube root of two, which is going to be the simplified version of the current expression. And with that being said, the answer to this problem is going to be d. So I hope this video helps you in understanding how to approach this problem and I'll go ahead and see you all in the next video.

Perform the indicated operations. Assume all variables represent positive real numbers. 2∛3 + 4∛24 - ∛81

Key Concepts

Radical Expressions

Simplifying Radicals

Combining Like Terms

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