What does a negative index mean?

A reciprocal: a^{-n}=\frac1{a^n}

Write a^{\frac mn} using roots.

\left(\sqrt[n]{a}\right)^m=\sqrt[n]{a^m}

Evaluate 27^{-\frac23}.

\frac{1}{(27^{\frac13})^2}=\frac19

An irrational root, such as \sqrt2, that cannot be written as a fraction of integers.

What are \sqrt{ab} and \sqrt{\frac ab} in terms of \sqrt a and \sqrt b?

\sqrt a\sqrt b and \frac{\sqrt a}{\sqrt b}

Is \sqrt{a+b}=\sqrt a+\sqrt b?

No. For example \sqrt{9+16}=5\ne7.

Rationalise \frac{1}{3-\sqrt5}.

Multiply by \frac{3+\sqrt5}{3+\sqrt5}: \frac{3+\sqrt5}{9-5}=\frac{3+\sqrt5}{4}

What is the conjugate of a+\sqrt b and what is the product of the pair?

a-\sqrt b; the product is a^2-b

Laws of indices and surdsEdexcel International A Level Maths: Flashcards

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Question

State the three laws of indices for $a^m$ and $a^n$.

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State the three laws of indices for $a^m$ and $a^n$ .: $a^m\times a^n=a^{m+n}$ , $\frac{a^m}{a^n}=a^{m-n}$ , $(a^m)^n=a^{mn}$
What is $a^0$ (for $a\ne0$ )?: $1$
What does a negative index mean?: A reciprocal: $a^{-n}=\frac1{a^n}$
Write $a^{\frac mn}$ using roots.: $\left(\sqrt[n]{a}\right)^m=\sqrt[n]{a^m}$
Evaluate $16^{\frac34}$ .: $(16^{\frac14})^3=2^3=8$
Evaluate $27^{-\frac23}$ .: $\frac{1}{(27^{\frac13})^2}=\frac19$
Solve $x^{\frac32}=216$ .: $x=216^{\frac23}=36$
What is a surd?: An irrational root, such as $\sqrt2$ , that cannot be written as a fraction of integers.
Simplify $\sqrt{75}$ .: $\sqrt{25\times3}=5\sqrt3$
What are $\sqrt{ab}$ and $\sqrt{\frac ab}$ in terms of $\sqrt a$ and $\sqrt b$ ?: $\sqrt a\sqrt b$ and $\frac{\sqrt a}{\sqrt b}$
Is $\sqrt{a+b}=\sqrt a+\sqrt b$ ?: No. For example $\sqrt{9+16}=5\ne7$ .
Rationalise $\frac{1}{3-\sqrt5}$ .: Multiply by $\frac{3+\sqrt5}{3+\sqrt5}$ : $\frac{3+\sqrt5}{9-5}=\frac{3+\sqrt5}{4}$
What is the conjugate of $a+\sqrt b$ and what is the product of the pair?: $a-\sqrt b$ ; the product is $a^2-b$

Exam questions on Laws of indices and surds

The number $N=16^{\frac34}\times27^{-\frac23}$ .
Hence find the exact value of $N^{-\frac12}$ , giving your answer in the form $k\sqrt2$ .2 marks
For $x>0$ , $f(x)=\dfrac{(2x^3)^2\sqrt{x}}{8x^5}$ .
Solve $f(x)=108$ .2 marks
A rectangle has area $(7+\sqrt5)$ cm $^2$ and length $(3-\sqrt5)$ cm.
Find the width of the rectangle, giving your answer in the form $\frac{p+q\sqrt5}{r}$ , where $p$ , $q$ and $r$ are integers.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).