3. (a) Find the value of $8^{\frac{1}{3}}$ (2) (b) Simplify fully $$\frac{\left(2x^{\frac{2}{3}}\right)^{3}}{4x^{-2}}$$ (3) - Edexcel - A-Level Maths Pure - Question 5 - 2013 - Paper 1

First, simplify the numerator:

$\left(2x^{\frac{2}{3}}\right)^{3} = 2^{3} \cdot \left(x^{\frac{2}{3}}\right)^{3} = 8x^{2}$

Now we rewrite the entire expression:

$\frac{8x^{2}}{4x^{-2}}$

Next, simplify the coefficients and use the properties of exponents:

$\frac{8}{4} \cdot \frac{x^{2}}{x^{-2}} = 2 \cdot x^{2 - (-2)} = 2 \cdot x^{2 + 2} = 2x^{4}$

Therefore, the fully simplified expression is:

$2x^{4}$