State the law for a^m\div a^n.

a^{m-n} (subtract the indices, same base).

What does a^{-n} mean?

\dfrac1{a^n}, the reciprocal of a^n. It is not a negative number.

What does a^{\frac1n} mean?

The nth root of a, \sqrt[n]{a}.

Write a^{\frac mn} using roots.

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

Evaluate 27^{\frac23}.

\left(\sqrt[3]{27}\right)^2=3^2=9

Evaluate \left(\dfrac49\right)^{-\frac12}.

\left(\dfrac94\right)^{\frac12}=\dfrac32

How do you solve x^{\frac32}=8?

Raise both sides to the power \frac23: x=8^{\frac23}=4.

Why is 3x^{-2} different from (3x)^{-2}?

3x^{-2}=\dfrac3{x^2} but (3x)^{-2}=\dfrac1{9x^2}: the index only applies to what is directly beneath it.

Laws of indicesAQA A-Level Maths: Flashcards

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State the law for $a^m\times a^n$.

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State the law for $a^m\times a^n$ .: $a^{m+n}$ (add the indices, same base).
State the law for $a^m\div a^n$ .: $a^{m-n}$ (subtract the indices, same base).
State the law for $(a^m)^n$ .: $a^{mn}$ (multiply the indices).
What is $a^0$ for $a\neq0$ ?: $1$
What does $a^{-n}$ mean?: $\dfrac1{a^n}$ , the reciprocal of $a^n$ . It is not a negative number.
What does $a^{\frac1n}$ mean?: The $n$ th root of $a$ , $\sqrt[n]{a}$ .
Write $a^{\frac mn}$ using roots.: $\left(\sqrt[n]{a}\right)^m$ or $\sqrt[n]{a^m}$ .
Evaluate $27^{\frac23}$ .: $\left(\sqrt[3]{27}\right)^2=3^2=9$
Evaluate $\left(\dfrac49\right)^{-\frac12}$ .: $\left(\dfrac94\right)^{\frac12}=\dfrac32$
Write $\dfrac{1}{\sqrt{x}}$ as a power of $x$ .: $x^{-\frac12}$
Simplify $\left(8x^6\right)^{\frac23}$ .: $4x^4$
How do you solve $x^{\frac32}=8$ ?: Raise both sides to the power $\frac23$ : $x=8^{\frac23}=4$ .
Why is $3x^{-2}$ different from $(3x)^{-2}$ ?: $3x^{-2}=\dfrac3{x^2}$ but $(3x)^{-2}=\dfrac1{9x^2}$ : the index only applies to what is directly beneath it.

Exam questions on Laws of indices

A student evaluates numerical powers without a calculator, using the laws of indices.
Evaluate $\left(\dfrac{27}{8}\right)^{-\frac23}$ , giving your answer as a fraction in its simplest form.2 marks
Throughout this question, $x>0$ .
Solve the equation $x^{-\frac32}=\dfrac18$ .2 marks
The function $f$ is defined by $f(x)=\dfrac{x^{2}+3x}{\sqrt{x}}$ for $x>0$ .
Write $f(x)$ in the form $x^{p}+kx^{q}$ , where $p$ , $q$ and $k$ are constants to be found.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).