Simplify each radical. Assume all variables represent positive re...

Question

Simplify each radical. Assume all variables represent positive real numbers. √24m⁶n⁵

Accepted Answer

Hey everyone here we are asked to simplify the following expression where we take the third root of 54 times A. To the 10th power times B to the seventh power. Now the key in simplifying this expression is to take the product underneath the root and find as many cubed terms from it that we can so we can begin with the coefficient here. Which 54? Well 54 can break up into two times three which gives us six times three giving us 18 times another three gives us 54. And so here we can see that 54 can become can break into two times three to the third power. And now we can move on to our age term A to the 10th power. Well again we're looking for cubed terms so we can break up A to the 10th power into a cubed times a cubed which gives us A to the six power times a cubed. Which gives us A to the ninth power. So then we just multiply by A. To the power of one and then we have A to the 10th power. And now we can move on to breaking up our B term where we have B to the seventh power. Well be to the seventh power can break up into B cubed times B cubed times B to the first power. So now we just need to rewrite our original expression with the expanded form of each of our terms here. So again we had the third root of times A to the 10th power times B to the seventh power where we see that the expression underneath the root expands. So we have the third root of two times three to the power of three times a times A. To the power of three times A. To the power of three and again times another A. To the power of three. And then we just have times A. To the first power which is just A. And next moving on to our B terms. We have times B to the third Power, times B to the third power. And lastly times B to the first power, which is just be. So now all we have to do is simplify by taking the third root of each of these terms underneath the root. So we can see that we cannot but simplify the third root of two A. Or B. So in our expression we will just have the third root of two A. B. But for the rest of our terms here we can take the third root. So our expression simplifies to three Simplifies to three times a times A. Times A times B times B. And then this is multiplied by the third root of two A. B. So now we just have to clean up our solution here and we see that our final simplified expression becomes three times A. To the third power times B squared times the third root of two A. B. And with this we have found our fully simplified expression and we are left with answer choice B. Thanks for watching. I hope you found this video helpful, and I'll see you again next time.

Simplify each radical. Assume all variables represent positive real numbers. √24m⁶n⁵

Key Concepts

Radical Simplification

Properties of Exponents

Combining Radicals

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