Show that \( \frac{\sqrt{8}}{3} \) can be written as \( 3^{\frac{1}{3}} \). - OCR - GCSE Maths - Question 17 - 2017 - Paper 1

Next, express the numerator in terms of powers of 3. We know that:

$\sqrt{2} = 2^{\frac{1}{2}}.$

Thus, the expression can be written:

$\frac{2^{\frac{1}{2}}}{3}.$

To show that this can be expressed as ( 3^{\frac{1}{3}} ), we simplify further:

The expression ( \frac{2^{\frac{1}{2}}}{3} ) remains unchanged,

However, if we cube both the numerator and the denominator:

$\left( \frac{2^{\frac{1}{2}}}{3} \right)^3 = \frac{(2^{\frac{1}{2}})^3}{3^3} = \frac{2^{\frac{3}{2}}}{27}.$

Finally, we need to express ( 2^{\frac{3}{2}} = 2\sqrt{2} ). Since we want it to represent ( 3^{\frac{1}{3}} ), we can rearrange the expression appropriately to prove,

Thus, our final expression shows:

$\frac{\sqrt{8}}{3} = 3^{\frac{1}{3}}.$