SOLUTION: Use rational exponents to write fourth root 2 * cube root 5 as a single radical expression.

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Question 234881: Use rational exponents to write fourth root 2 * cube root 5 as a single radical expression.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
root%284%2C+2%29%2Aroot%283%2C+5%29
To combine these into a single radical, we will need the "roots" to match. By "roots" I mean the index on the radical. Your "roots" are currently 4 and 3.

To change the roots we will rewrite this expression using rational exponents. Since root%284%2C+x%29+=+x%5E%281%2F4%29 and root%283%2C+x%29+=+x%5E%281%2F3%29 we have:
2%5E%281%2F4%29+%2A5%5E%281%2F3%29
Now that our "roots" are denominators of fractions, we can match them just like we do when we are adding fractions. (We are not actually going to add these fractions but we are going to match the denominators as if we were.) The Lowest Common Denominator is 12. So we can rewrite the fractions with denominators of 12:
2%5E%283%2F12%29+%2A5%5E%284%2F12%29
Now that our "roots" are the same, let's go back to radical form, using the fact that x%5E%28a%2Fb%29+=+root%28b%2C+x%5Ea%29:
root%2812%2C+2%5E3%29%2Aroot%2812%2C+5%5E4%29
And with our matching roots, we can now use the property root%28a%2C+x%29%2Aroot%28a%2C+y%29+=+root%28a%2C+x%2Ay%29 to multiply:
root%2812%2C+2%5E3%2A5%5E4%29
Since 2%5E3+=+8 and 5%5E4+=+625 and 8%2A625+=+5000, this simplifies to:
root%2812%2C+5000%29