12 (a) Find the value of $81^{rac{1}{2}}$ - Edexcel - GCSE Maths - Question 12 - 2017 - Paper 1

To evaluate $\left( \frac{64}{125} \right)^{\frac{3}{2}}$, we will first find the square root of both the numerator and the denominator:

$\left( \frac{64}{125} \right)^{\frac{3}{2}} = \frac{(\sqrt{64})^3}{(\sqrt{125})^3}.$

Now, calculating each square root:

• $\sqrt{64} = 8$, so $\left( \sqrt{64} \right)^3 = 8^3 = 512$.
• $\sqrt{125} = 5\sqrt{5}$, so $\left( \sqrt{125} \right)^3 = (5\sqrt{5})^3 = 125 * 5\sqrt{5} = 625\sqrt{5}$.

Putting it all together: $\left( \frac{64}{125} \right)^{\frac{3}{2}} = \frac{512}{625\sqrt{5}}.$

After simplification, if needed, or if left as is, the answer could also be approximated. However, simplifying is not necessary initially.