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

Question

Perform the indicated operations. Assume all variables represent positive real numbers. ∜81x⁶y³ - ∜16x¹⁰y³

Accepted Answer

Hello everyone. We're going to simplify the expression. The fourth root of 625 seat to the sixth D. To the third minus the fourth root of 256. C. To the 10th D. To the third. I'm gonna simplify each term separately and then simplify the expression as a whole. So the first term I'm simplifying is the fourth root of 625. C. To the sixth D. To the third. I'm gonna break this down. I'm gonna rewrite it as the 4th root Of because that's my constant. And I know I'm going to be able to get an actual number from it times my variable of C. To the sixth. And I'm going to raise this to the fourth power because that's what radical four is as an exponent time's D. To the third and again it's raised to the fourth power. So to the 1/4. Now simplifying this. The 4th root of 625 is five. Recalling that when I raise the power to a power I multiply my exponents. We get seed the 6/4 times D. To the 3/4. And there's one more thing I can do to simplify this before I put it back into a radical form and that is my exponent on C. Is an improper fraction. This means I can break it down I can take out a factor of one. So see to the 4/4 time c. two. The 2/4 is equivalent to what we had My d. to the 3 4th just comes down around. So see to the 4/4 is the same thing as saying see the first which is the same as C. And then I'm gonna keep times C. To the 2/4 times D. To the 34. I recall since both denominators of my exponents are still the same. That means they're still under this a radical with the same index in this case. four. So I'm going to rewrite this with a radical the five c. don't have an exponent. So they go on the outside of the radical inside the fourth root we're going to have C. Squared D. To the third. So now jumping to my other term. So the term of the 4th root Of 256. c. to the 10th. D. to the third. I'm going to go through the same process. So first I'm gonna break it apart And I'm gonna do the 4th root of 256 Times. See the 10th. And I'm raising it to the 1/4 because I want to get it out of radical form and into exponential form a little bit. And same with D. So D. To the third to the 1/4. The 4th root of our constant. So 256 is four C. to the 10th to the 1/4 gives us c. to the 10/4. And then D. To the third to the 1/4 is D. To the 3/4. So very similar to what I did in the first term. I'm going to break down this C. Term this time I'm gonna break it down into three different parts because I want to take factors of one out So I can do c. to the 4/4. I still have six here times another c. to the 4/4s times what's left of C. Two, the 2/4. Remember when you multiply you add your exponents. So if I'm pulling it apart, enumerators here still add up to 10 And then deed the 3/ isn't changing. so 4/4 is the same as one. So this is now see to the I apologize four times C. Times C. Times C. To the 2/4 times D. To the 3/4. Now right here C. Times C. Is the same as C squared. So we have four C. Squared Times c. to the 2/4 times d. to the 3/4. And because we still have the same denominator on our exponents we're gonna put this back into radical form. So on the outside of my radical is four C squared and under the radical with the index of four I have C squared D. To the third. We're almost there. So now that I have all these pieces I'm gonna put them together. I need to subtract these from each other. So I have five C. Times the fourth root of C squared D. To the third minus four C squared times the fourth root of C squared D. To the third. And I'm going to put parentheses around my radicals, just so we know where the index goes if you notice are radicals and the things under them are exactly the same. This means we can factor this out. So when we factor it out we get five c minus four C squared times the fourth route of C squared D to the third. And there's nothing else we can do there. This is our final answer, which when we look back is answer choice A Have a good day.

Perform the indicated operations. Assume all variables represent positive real numbers. ∜81x⁶y³ - ∜16x¹⁰y³

Key Concepts

Radical Expressions

Properties of Exponents

Simplifying Expressions

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