Simplify \( a^3 \times a^5 \).
\( a^8 \).
Indices & Surds
10 exam-style questions on laws of indices, negative and fractional indices, and simplifying surds.
4 Easy 4 Medium 2 Hard
Simplify \( \dfrac{x^7}{x^2} \).
\( x^5 \).
Evaluate \( 2^{-3} \).
\( \dfrac{1}{2^3} = \dfrac{1}{8} \).
Simplify \( \sqrt{50} \).
\( \sqrt{25 \times 2} = 5\sqrt{2} \).
Simplify \( (3x^2)^3 \).
\( 27x^6 \).
Evaluate \( 16^{3/4} \).
\( (16^{1/4})^3 = 2^3 = 8 \), or \( (16^3)^{1/4} \).
Simplify \( \sqrt{18} + \sqrt{8} \).
\( 3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2} \).
Rationalise the denominator of \( \dfrac{5}{\sqrt{2}} \).
\( \dfrac{5}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{5\sqrt{2}}{2} \).
Simplify \( \dfrac{6}{\sqrt{3} + 1} \) by rationalising the denominator.
Multiply by \( \sqrt{3}-1 \): \( \dfrac{6(\sqrt{3}-1)}{3-1} = \dfrac{6(\sqrt{3}-1)}{2} = 3(\sqrt{3}-1) \).
Simplify \( (2\sqrt{3} + 1)(2\sqrt{3} - 1) \).
Difference of squares: \( (2\sqrt{3})^2 - 1^2 = 12 - 1 = 11 \).