Simplify \(\left(8x^6\right)^{\frac{2}{3}}\), giving your answer in the form \(ax^n\).
Question 1 Report
Simplify \(\left(8x^6\right)^{\frac{2}{3}}\), giving your answer in the form \(ax^n\).
Answer Details
An index outside a bracket applies to every factor inside, and a fractional index means root then power. Deal with the number and the letter separately.
For the number: \(8^{\frac{2}{3}}\) means the cube root of 8, which is 2, then squared.
\(8^{\frac{2}{3}} = 4\) [B1]
For the letter, a power of a power multiplies the indices: \(\left(x^6\right)^{\frac{2}{3}} = x^{6 \times \frac{2}{3}} = x^4\). Combining the two parts:
\(4x^4\) [B1]
The answer is in the required form \(ax^n\) with \(a = 4\) and \(n = 4\).
The error that costs the first mark is leaving the coefficient alone and writing \(8x^4\); the outer index applies to the 8 just as much as to \(x^6\). The second trap is multiplying \(6 \times \frac{3}{2} = 9\) by flipping the fraction; the index is multiplied as it stands, and dividing by 3 then multiplying by 2 gives 4.
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