Question

factor the following expression? Do not leave negative exponents.
21(x−8)^1/4 (x^2+7)^2/3 +14(x−8)^5/4(x^2+7)^−1/3

Answer 1

You can see there is (x-8) and (x^2+27) in both terms.
You probably want to factor out the lowest power of those factors.
also, 21 and 14 are both multiples of 7, so
21(x−8)^1/4 (x^2+7)^2/3 +14(x−8)^5/4(x^2+7)^−1/3
7(x-8)^1/4 (x^2+7)^-1/3 (3(x^2+7) + 2(x-8))
7(x-8)^1/4 / (x^2+7)^1/3 (3x^2+21+2x-16)

7(x-8)^1/4 (3x^2+2x-5)
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(x^2+7)^1/3

or

7(x-8)^1/4 (x-1)(3x+5)
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(x^2+7)^1/3

Answer 2

oops. 21-16 = 5, not -5

7(x-8)^1/4 (3x^2+2x+5)
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(x^2+7)^1/3