TMUA Indices and Surds

Index laws

• $a^m\cdot a^n=a^{m+n}$ and $a^m\div a^n=a^{m-n}$.
• $(a^m)^n=a^{mn}$.
• $a^{-n}=\dfrac1{a^n}$ and $a^{m/n}=\left(\sqrt[n]{a}\right)^m$.

Simplifying surds

Pull out square factors: $\sqrt{12}=2\sqrt3$. Only like surds can be added, so $2\sqrt3+3\sqrt3=5\sqrt3$.

Rationalising

To clear a surd from a denominator $p+\sqrt q$, multiply top and bottom by $p-\sqrt q$, using $(p+\sqrt q)(p-\sqrt q)=p^2-q$.

Working without a calculator

Know your squares up to $15^2$, cubes up to $5^3$ and powers of 2 up to $2^{10}$. They let you spot exact roots at a glance, such as $\sqrt{196}=14$ or $\sqrt[3]{125}=5$.

When an equation has different bases, rewrite them with a common base. For example $4^x=8^{x-1}$ becomes $2^{2x}=2^{3x-3}$, so $x=3$. Fractional indices combine two ideas: $16^{3/4}=\left(\sqrt[4]{16}\right)^3=2^3=8$.

In surd questions, expand carefully and look for cancellation, because products of conjugates collapse to whole numbers.

Always simplify at the end. An option written as $\dfrac{5\sqrt3}{3}$ and one written as $\dfrac{5}{\sqrt3}$ are equal in value, so reduce your answer to a standard form and compare it with the options carefully. Negative indices on a fraction flip it, and a zero index gives 1 for any non-zero base.

Check by estimating: $\sqrt{50}$ must be a little over 7, so $5\sqrt2$ is plausible.

Worked examples

Common mistakes

• Writing $(a+b)^2=a^2+b^2$ and missing the cross term, especially with surds.
• Adding unlike surds, such as $\sqrt2+\sqrt3=\sqrt5$.
• Applying a negative power to the sign instead of taking the reciprocal.
• Rationalising with the same sign instead of the conjugate.

Test yourself on this topic with our free practice questions.