Asked by cvf
What is the simplified form of the following expression? Assume x greater-than 0
RootIndex 4 StartRoot StartFraction 3 Over 2 x EndRoot EndFraction
StartFraction RootIndex 4 StartRoot 6 x EndRoot Over 2 x EndFraction
StartFraction RootIndex 4 StartRoot 24 x cubed EndRoot Over 2 x EndFraction
StartFraction RootIndex 4 StartRoot 24 x cubed EndRoot Over 16 x Superscript 4 Baseline EndFraction
RootIndex 4 StartRoot 12 x squared EndRoot
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Answered by
GPT-5 mini
AI
Interpret the expression as the product
4th root(3/(2x)) * [4th root(6x)/(2x)] * [4th root(24 x^3)/(2x)] * [4th root(24 x^3)/(16 x^4)] * 4th root(12 x^2).
Combine the fourth roots into one and multiply the external denominators:
= 4th root( (3/(2x))·(6x)·(24x^3)·(24x^3)·(12x^2) ) / (2x·2x·16x^4)
= 4th root(62208 x^8) / (64 x^6).
Since x>0, 4th root(x^8)=x^2, and 62208 = 12^4·3, so 4th root(62208)=12·4th root(3). Thus
= (12·4th root(3)·x^2) / (64 x^6) = (3·4th root(3)) / (16 x^4).
Final answer: (3 * fourth-root(3)) / (16 x^4).
4th root(3/(2x)) * [4th root(6x)/(2x)] * [4th root(24 x^3)/(2x)] * [4th root(24 x^3)/(16 x^4)] * 4th root(12 x^2).
Combine the fourth roots into one and multiply the external denominators:
= 4th root( (3/(2x))·(6x)·(24x^3)·(24x^3)·(12x^2) ) / (2x·2x·16x^4)
= 4th root(62208 x^8) / (64 x^6).
Since x>0, 4th root(x^8)=x^2, and 62208 = 12^4·3, so 4th root(62208)=12·4th root(3). Thus
= (12·4th root(3)·x^2) / (64 x^6) = (3·4th root(3)) / (16 x^4).
Final answer: (3 * fourth-root(3)) / (16 x^4).
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