FN3, FN7, HN3, HN7 - Laws of indices | Eduqas GCSE Mathematics

Powers can make a calculation look enormous even when its answer is small. Learn to combine integer powers, explain zero and negative indices, and simplify powers of powers before calculating. FN7 and HN7 supply the index laws; FN3 and HN3 connect negative powers to reciprocals and division.

Products and quotients

Can you work out $2^6\times2^5\div2^8$ without finding any of the three large powers? The indices count factors, so we can count and cancel first.

You already know that $2^3=2\times2\times2$ . Multiplying $2^3$ by $2^2$ gives five factors of $2$ , so $2^3\times2^2=2^5$ . Dividing $2^5$ by $2^2$ cancels two of those factors, leaving $2^3$ .

Base and index

In $a^n$ , $a$ is the base and $n$ is the index, also called the exponent. For a positive integer $n$ , the power means a product of $n$ equal factors.

The base stays the same when we combine equal factors. The index changes because the number of factors changes.

Same-base laws

a^m\times a^n=a^{m+n},\qquad a^m\div a^n=a^{m-n}.

The bases must match. In division, $a\ne0$ . We first use positive integer indices; the next part extends their meaning to zero and negative integers.

Not in the formula sheets: learn these laws.

For example, $2^3\times3^2$ has different bases, so adding the indices would not count equal factors.

Worked example 1: combine before calculating

Question. Work out $2^4\times2^3\div2^2$ .

Plan. Add the indices for multiplication, then subtract the divisor's index.

\begin{aligned} 2^4\times2^3\div2^2&=2^{4+3-2} &&\text{count the remaining factors}\\ &=2^5 &&\text{simplify the index}\\ &=32 &&\text{evaluate the power} \end{aligned}

Answer. $32$ .

Check. Direct calculation gives $16\times8\div4=128\div4=32$ .

Keep the structure, but increase the power in the denominator. More factors will cancel.

Worked example 2: more cancellation

Question. Work out $2^4\times2^3\div2^6$ .

Plan. Combine the same-base powers before evaluating.

\begin{aligned} 2^4\times2^3\div2^6&=2^{4+3-6} &&\text{subtract the divisor's index}\\ &=2^1 &&\text{one factor remains}\\ &=2 \end{aligned}

Answer. $2$ .

Check. The numerator is $128$ and the denominator is $64$ ; $128\div64=2$ .

The operation between powers decides the rule. An addition sign does not join their factors.

Watch out

$2^3+2^2\ne2^5$ : the left side is $8+4=12$ . Add indices when multiplying powers with the same base, not when adding their values.

Use the same cancellation idea with a different base.

Zero and negative indices

Reducing an index by one divides the value by its base. Continue the pattern below instead of interpreting a negative index as a negative answer.

Power	Value
$2^2$	$4$
$2^1$	$2$
$2^0$	$1$
$2^{-1}$	$\frac12$
$2^{-2}$	$\frac14$

There is also a division explanation. $2^3\div2^3=1$ , while subtracting the indices gives $2^{3-3}=2^0$ . And $2^2\div2^5=1/2^3$ , while subtracting the indices gives $2^{-3}$ . Both descriptions must give the same value.

Zero and negative powers

For $a\ne0$ and a positive integer $n$ ,

a^0=1,\qquad a^{-n}=\frac1{a^n}.

A negative index means take the reciprocal. Zero cannot be used as the base in these rules because division by zero is undefined. We do not assign a value to $0^0$ here; $0^n=0$ when $n$ is a positive integer.

Not in the formula sheets: learn these rules.

The minus sign in an index has a different job from a minus sign in the base. With a negative base, use brackets to show that the sign is part of each repeated factor.

Worked example 3: a reciprocal power

Question. Work out $3^0+3^{-3}$ . Give your answer as a fraction.

Plan. Replace the zero power by $1$ , and the negative power by a reciprocal.

\begin{aligned} 3^0+3^{-3}&=1+\frac1{3^3} &&\text{use the two rules}\\ &=1+\frac1{27} &&\text{evaluate the denominator}\\ &=\frac{28}{27} &&\text{add equal-denominator fractions} \end{aligned}

Answer. $\frac{28}{27}$ .

Check. $3^{-3}$ must multiply $27$ to give $1$ , so it is a small positive amount. The answer is just above $1$ , as expected.

Now change only the base to $-3$ . The reciprocal rule still works, but the odd power keeps the negative sign.

Worked example 4: a negative base

Question. Work out $(-3)^0+(-3)^{-3}$ . Give your answer as a fraction.

Plan. Keep the brackets when finding the reciprocal power.

\begin{aligned} (-3)^0+(-3)^{-3}&=1+\frac1{(-3)^3} &&\text{the base is nonzero}\\ &=1-\frac1{27} &&\text{three negative factors give }-27\\ &=\frac{26}{27} \end{aligned}

Answer. $\frac{26}{27}$ .

Check. $(-27)\times(-1/27)=1$ , confirming the reciprocal. Adding that negative reciprocal to $1$ gives a value just below $1$ .

With an even power, the sign changes differently: $(-3)^{-2}=1/9$ , whereas $-3^{-2}=-(1/9)$ . In the second expression the base is $3$ ; the outside minus sign is applied afterwards.

Watch out

$5^{-2}$ is $1/25$ , not $-25$ and not $1/10$ . A negative index asks for a reciprocal; a power still means repeated multiplication.

Keep the negative base inside its brackets throughout this calculation.

Powers of powers

A power outside brackets repeats everything inside them. For example, $(2^3)^2=2^3\times2^3=2^6$ : two groups of three equal factors give six factors in all.

Similarly, $(2\times3)^2=(2\times3)(2\times3)=2^2\times3^2$ . Each factor inside the brackets is squared. A quotient works in the same way because each repeated fraction contributes a numerator and a denominator.

Powers of powers, products and quotients

For integer indices wherever the expressions are defined,

(a^m)^n=a^{mn},\qquad (ab)^n=a^nb^n,\qquad \left(\frac ab\right)^n=\frac{a^n}{b^n}.

The quotient needs $b\ne0$ . Any base raised to a zero or negative index must also be nonzero.

Not in the formula sheets: learn these laws.

Multiply the indices in a power of a power; do not add them. We can use this rule to avoid large intermediate calculations.

Worked example 5: repeated products

Question. Work out $\dfrac{((2\times3)^2)^2}{3^4}$ .

Plan. Combine the outside powers, then split the product so the powers of $3$ cancel.

\begin{aligned} \frac{((2\times3)^2)^2}{3^4} &=\frac{(2\times3)^4}{3^4} &&\text{multiply the indices}\\ &=\frac{2^4\times3^4}{3^4} &&\text{raise each factor}\\ &=2^4=16 &&\text{cancel the nonzero factor} \end{aligned}

Answer. $16$ .

Check. The original numerator is $36^2=1296$ . Since $81\times16=1296$ , division by $3^4=81$ gives $16$ .

Change the product inside the brackets to a quotient. Now the power of $3$ starts in the denominator.

Worked example 6: repeated quotients

Question. Work out $\left(\left(\frac23\right)^2\right)^2\times3^4$ .

Plan. Combine the powers, raise numerator and denominator, then cancel.

\begin{aligned} \left(\left(\frac23\right)^2\right)^2\times3^4 &=\left(\frac23\right)^4\times3^4 &&\text{multiply the indices}\\ &=\frac{2^4}{3^4}\times3^4 &&\text{raise both parts of the fraction}\\ &=2^4=16 &&\text{cancel equal factors} \end{aligned}

Answer. $16$ .

Check. Squaring $4/9$ gives $16/81$ . Multiplying by $81$ restores $16$ .

Division by a negative power also uses the same subtraction law: $5^{-2}\div5^{-5}=5^{-2-(-5)}=5^3=125$ . Subtract the whole signed index. The opening calculation is now short: $2^6\times2^5\div2^8=2^{6+5-8}=2^3=8$ .

Watch out

$(2^3)^2=2^6$ , but $2^3\times2^2=2^5$ . The brackets tell you whether you have a power of a power or a product of two powers.

Apply the quotient rule to another fraction before evaluating.

Quick Recap

Multiply same-base powers by adding their indices; divide by subtracting the divisor's index.
A nonzero number to power zero is $1$ ; a negative index means take a reciprocal.
Raise a power to a power by multiplying the indices.
A power of a product or quotient applies to every factor.

The opening expression equals $8$ : simplify the indices before calculating the powers.

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FN3, FN7, HN3, HN7 - Laws of indices

Products and quotients

Zero and negative indices

Powers of powers

Quick Recap

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