Powers and roots: Practice Questions

Evaluate the expression \(\left(\frac{125}{64}\right)^{-\frac{2}{3}}\).
Give your answer as a fraction in its simplest form.

Simplify the following expression involving surds fully:
\(\frac{\sqrt{50} - \sqrt{18}}{\sqrt{2}} + \sqrt{12} \times \sqrt{3}\)

Calculate the value of \(64^{\frac{2}{3}}\).

Solve the following equation for \(x\):
\(4^{2x-1} = 32\)

Work out the exact value of \( 16^{-\frac{3}{4}} \).

Simplify the following expression using index laws:
\(x^5 \times x^3\) and \(y^{12} \div y^4\).

Work out the value of \( \left(\frac{1}{64}\right)^{-\frac{1}{3}} \).

Given that \(y = 5\sqrt{3} - 2\), calculate the value of \(y^2\).
Give your answer in the form \(a + b\sqrt{3}\), where a and b are integers.

(a) Simplify fully \(\frac{15a^7b^2}{3a^3b^5}\), giving your answer with positive indices.

(b) Show that \(\frac{4}{2 + \sqrt{3}}\) can be written in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers.

(c) Evaluate \(\left(\frac{64}{125}\right)^{-\frac{2}{3}}\).

(a) Simplify fully \( \frac{(3x^2y^3)^2}{9x^3y} \).

(b) Show that \( \frac{12}{3 - \sqrt{5}} \) can be written in the form \( a + b\sqrt{5} \), where \( a \) and \( b \) are integers.

(c) Solve the equation \( 27^{x-2} = 9^{x+4} \).