2. (a) Write down the value of $16^{\frac{1}{4}}$ - Edexcel - A-Level Maths Pure - Question 4 - 2008 - Paper 2

We start by applying the power of a product rule, which states that $(ab)^n = a^n b^n$. Therefore:

$(16x^{12})^{\frac{3}{4}} = 16^{\frac{3}{4}} \cdot (x^{12})^{\frac{3}{4}}.$

First, we calculate $16^{\frac{3}{4}}$:

1. We know $16 = 2^4$, so:

$16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8.$

2. Now for $(x^{12})^{\frac{3}{4}}$:

$(x^{12})^{\frac{3}{4}} = x^{12 \cdot \frac{3}{4}} = x^9.$

Combining both results gives:

$8x^9.$