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
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
oops. 21-16 = 5, not -5
7(x-8)^1/4 (3x^2+2x+5)
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(x^2+7)^1/3
To factor the given expression, we can first factor out the common terms from both terms of the expression.
The common factors are (x−8)^1/4 and (x^2+7)^1/3.
Factoring out the common factors, we have:
21(x−8)^1/4 (x^2+7)^2/3 + 14(x−8)^5/4(x^2+7)^−1/3
= (x−8)^1/4 (x^2+7)^1/3 [21(x^2+7) + 14(x^−8)^5/4]
Now, let's simplify the expression within the brackets:
= (x−8)^1/4 (x^2+7)^1/3 [21x^2+147 + 14(x−8)^5/4]
To simplify further, we can expand (x−8)^5/4 as (x−8)(x−8)(x−8)(x−8)(x−8)^1/4 and simplify (x^−8)^1/4 as √(x−8), and distribute the factors:
= (x−8)^1/4 (x^2+7)^1/3 [21x^2+147 + 14(x−8)(x−8)(x−8)(x−8)√(x−8)]
= (x−8)^1/4 (x^2+7)^1/3 [21x^2+147 +14(x^5−40x^4+480x^3−2560x^2+4096)√(x−8)]
Now, we have factored the given expression by factoring out the common factors.