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

Question

Simplify each radical. Assume all variables represent positive real numbers. ∛(16 (-2)⁴ (2)⁸)

Accepted Answer

Hey everyone here we are asked to simplify the following expression of the third root of 81 times the quantity of negative three raised to the sixth power times the quantity of three raised to the 10th power. So now the key in simplifying this expression here is to take each value that's underneath the root and rewriting it as as many p cubed terms as we can. So we can take the first value here which is 81. Well 81 is the same as three times three times three times three. And this is equivalent to three to the power of four which we know is the same as three times three cubed. So now we can move on to our next value here where we have the quantity of negative three raised to the sixth power. While we can break up this six power by rewriting this term as the quantity of negative three cubed times the quantity of negative three cubed. Now we can move on to our final value here underneath the root which is three raised to the 10th power. Well the 10th power can be bro broken up into multiple cubed terms. So we have three cubed times three cubed, times three cubed and finally times three to the first power. So now the next step is to just rewrite our original expression using these expanded forms. So our original expression becomes the third roots of three times three to the power of three times the quantity of negative three raised to the power of three times the quantity of negative three raised to the power of three as well. Then we have times three raised to the power of three times three to the power of three times another three to the power of three and finally times three to the power of one which is just three. So now that we have rewritten the values underneath the root as they're fully expanded form here we can go ahead and start taking the cubed root of each of these values. So doing this. Our expression now becomes three times negative three times negative three times three times three times three. And then we multiply each of these terms by the third root of three times three because those are the last two values left in the root. That cannot be further simplified. So now we do Just have to clean up our expression here by multiplying all of these terms together. So doing this outside of the root we are left with 729 and then this value is multiplied by the cubed root of nine. And with this we have found our fully simplified expression and so we are left with answer choice D. 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. ∛(16 (-2)⁴ (2)⁸)

Key Concepts

Radical Expressions

Exponents and Powers

Properties of Radicals

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