Laws of indices for all rational exponents, surd manipulation and rationalising denominators, and the meaning of negative and fractional indices.

What this dot point is asking

OCR wants you to apply the laws of indices for all rational exponents, interpret negative and fractional powers, simplify and combine surds, and rationalise denominators, including denominators of the form $a + b\sqrt{c}$ using the conjugate. These are non-calculator skills that underpin algebra throughout the course.

The answer

The laws of indices

For any non-zero base and rational powers:

A fractional power $a^{m/n}$ means "take the $n$th root, then raise to the $m$th power" (the order does not matter, but rooting first usually keeps numbers small). A negative power means "take the reciprocal".

Surds

A surd is an irrational root such as $\sqrt{2}$ or $\sqrt[3]{5}$. The key manipulation rules are $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ and $\sqrt{\tfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}$. To simplify a surd, take out the largest square factor: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$.

You can add or subtract only like surds: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$, but $\sqrt{2} + \sqrt{3}$ cannot be combined.

Rationalising the denominator

A fraction is "rationalised" when no surd appears in the denominator. For a single surd, multiply top and bottom by that surd. For a denominator $a + b\sqrt{c}$, multiply by the conjugate $a - b\sqrt{c}$, because $(a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - b^2 c$ is rational.

Examples in context

Combining the skills

Index and surd work appears inside almost every algebra question, for instance simplifying a derivative written with fractional powers, or tidying an answer so a marker can read it.

Solving an equation with a hidden quadratic in a power

Index laws let you spot a "hidden quadratic" when an unknown appears in an exponent, for example $4^x - 5(2^x) + 4 = 0$. Writing $y = 2^x$ turns $4^x = (2^x)^2 = y^2$, so the equation becomes the quadratic $y^2 - 5y + 4 = 0$. This substitution trick recurs in the exponentials topic too.

Why exact form matters

OCR frequently demands answers "in exact form" or "in the form $a + b\sqrt{c}$". A decimal from a calculator would lose marks where an exact surd or fraction is required, so rationalising and simplifying surds is not optional tidying but a graded skill. Leaving a surd in a denominator, or a decimal where a fraction is asked for, is treated as an incomplete answer.

Try this

Q1. Simplify $\sqrt{50} + \sqrt{18}$. [2 marks]

• Cue. $5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$.

Q2. Rationalise and simplify $\dfrac{6}{\sqrt{3}}$. [2 marks]

• Cue. Multiply by $\sqrt{3}/\sqrt{3}$ to get $\dfrac{6\sqrt{3}}{3} = 2\sqrt{3}$.