Indices and Roots

This lesson covers the laws of indices, the meaning of a zero index and of negative indices, square and cube roots, and — for Higher tier [H] — fractional indices. Index notation is a compact way of writing repeated multiplication, and the index laws it obeys run right through the OCR GCSE Mathematics (J560) Number and Algebra strands, including standard form and surds later in this course.

This lesson mainly builds AO1 fluency in applying the index laws, with AO2 reasoning when you explain why $a^{0} = 1$a0=1, and AO3 problem-solving in the multi-step evaluations.

Key Vocabulary

Term	Meaning
Base	The number being multiplied repeatedly, the $a$a in $a^{n}$an
Index (power, exponent)	How many times the base is multiplied by itself
Square root	A number that multiplies by itself to give the original; $\sqrt{25} = 5$25​=5
Cube root	A number cubed to give the original; $\sqrt[3]{27} = 3$327​=3
Reciprocal	$a^{-1} = \dfrac{1}{a}$a−1=a1​
Index law	A rule for combining powers, e.g. $a^{m} \times a^{n} = a^{m+n}$am×an=am+n

What an Index Means

Before the rules, be clear on what the notation says. The expression $a^{n}$an means "$a$a multiplied by itself $n$n times". So $2^{4}$24 means $2 \times 2 \times 2 \times 2 = 16$2×2×2×2=16, and $5^{3}$53 means $5 \times 5 \times 5 = 125$5×5×5=125. The number $a$a is the base and $n$n is the index (also called the power or exponent). Reading an index correctly is half the battle: $3^{2}$32 is "three squared" $= 9$=9, not "three times two" $= 6$=6 — a slip that loses easy marks on Paper 1.

Once you see indices as shorthand for repeated multiplication, the index laws stop being something to memorise blindly and become obvious. For example, $2^{3} \times 2^{4}$23×24 is $(2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2)$(2×2×2)×(2×2×2×2), which is $2$2 multiplied by itself $3 + 4 = 7$3+4=7 times, i.e. $2^{7}$27. That is exactly the multiplication law below.

The Laws of Indices

When the base is the same, powers combine by simple rules. Each one comes directly from counting how many times the base is multiplied.

Law	Rule	Example
Multiplication	$a^{m} \times a^{n} = a^{m+n}$am×an=am+n	$2^{3} \times 2^{4} = 2^{7}$23×24=27
Division	$a^{m} \div a^{n} = a^{m-n}$am÷an=am−n	$5^{6} \div 5^{2} = 5^{4}$56÷52=54
Power of a power	$(a^{m})^{n} = a^{mn}$(am)n=amn	$(3^{2})^{4} = 3^{8}$(32)4=38

The division law works the same way in reverse: $\dfrac{5^{6}}{5^{2}}$5256​ has six $5$5s on top and two on the bottom, so two cancel and four remain, giving $5^{4}$54. The power-of-a-power law comes from "$(3^{2})$(32) four times", which is $3^{2} \times 3^{2} \times 3^{2} \times 3^{2} = 3^{2+2+2+2} = 3^{8}$32×32×32×32=32+2+2+2=38 — so you multiply the indices.

OCR Exam Tip: The index laws only apply when the base is the same. You cannot simplify $2^{3} \times 5^{2}$23×52 with these rules because the bases ($2$2 and $5$5) differ — you would just work it out as $8 \times 25 = 200$8×25=200.

Worked Example 1

Simplify $a^{5} \times a^{3} \div a^{2}$a5×a3÷a2.

Add then subtract the indices: $a^{5+3-2} = a^{6}$a5+3−2=a6.

Answer: $a^{6}$a6.

Worked Example 2

Simplify $\dfrac{6x^{5} \times 2x^{4}}{4x^{3}}$4x36x5×2x4​.

Numbers: $\dfrac{6 \times 2}{4} = 3$46×2​=3. Powers of $x$x: $x^{5+4-3} = x^{6}$x5+4−3=x6.

Answer: $3x^{6}$3x6.

Common error: Multiplying the indices in $a^{5} \times a^{3}$a5×a3 to get $a^{15}$a15. You add indices when multiplying powers of the same base.

The Zero Index and Negative Indices

Following the division law, $a^{n} \div a^{n} = a^{0}$an÷an=a0; but anything divided by itself is $1$1, so:

$a^{0} = 1 \quad (\text{for } a \ne 0).$a0=1(for a=0).

A negative index means "reciprocal":

$a^{-n} = \dfrac{1}{a^{n}}.$a−n=an1​.

Worked Example 3

Work out (a) $7^{0}$70, (b) $3^{-2}$3−2, (c) $\left(\dfrac{2}{5}\right)^{-1}$(52​)−1.

(a) $7^{0} = 1$70=1. (b) $3^{-2} = \dfrac{1}{3^{2}} = \dfrac{1}{9}$3−2=321​=91​. (c) The reciprocal of $\dfrac{2}{5}$52​ is $\dfrac{5}{2}$25​.

Answers: (a) $1$1, (b) $\dfrac{1}{9}$91​, (c) $\dfrac{5}{2}$25​.

Common error: Writing $3^{-2} = -9$3−2=−9 or $-6$−6. A negative index does not make the answer negative; it makes a reciprocal.

Square and Cube Roots

A square root undoes squaring; a cube root undoes cubing. Knowing your squares and cubes makes these instant on the non-calculator paper.

$n$n	$n^{2}$n2	$n^{3}$n3
$2$2	$4$4	$8$8
$3$3	$9$9	$27$27
$4$4	$16$16	$64$64
$5$5	$25$25	$125$125

Worked Example 4

Work out (a) $\sqrt{81}$81​, (b) $\sqrt[3]{64}$364​, (c) $\sqrt{144}$144​.

(a) $9 \times 9 = 81$9×9=81, so $\sqrt{81} = 9$81​=9. (b) $4^{3} = 64$43=64, so $\sqrt[3]{64} = 4$364​=4. (c) $12^{2} = 144$122=144, so $\sqrt{144} = 12$144​=12.

Answers: (a) $9$9, (b) $4$4, (c) $12$12.

Worked Example 5

Work out $\sqrt{36} + \sqrt[3]{27} \times 2$36​+327​×2.

By BIDMAS, roots act like indices (do them first), then multiply, then add: $\sqrt{36} = 6$36​=6, $\sqrt[3]{27} = 3$327​=3, so $6 + 3 \times 2 = 6 + 6 = 12$6+3×2=6+6=12.

Answer: $12$12.

Worked Example 5b

Write $\sqrt{49}$49​ and $\sqrt[3]{125}$3125​ as products, then work out $\sqrt{49} \times \sqrt[3]{125}$49​×3125​.

$\sqrt{49} = 7$49​=7 (since $7^{2} = 49$72=49) and $\sqrt[3]{125} = 5$3125​=5 (since $5^{3} = 125$53=125). So $\sqrt{49} \times \sqrt[3]{125} = 7 \times 5 = 35$49​×3125​=7×5=35.

Answer: $35$35. Knowing the common squares ($1, 4, 9, 16, 25, 36, 49, 64, 81, 100$1,4,9,16,25,36,49,64,81,100) and cubes ($1, 8, 27, 64, 125$1,8,27,64,125) by heart makes root questions instant on the non-calculator paper.

Fractional Indices [H]

This section is Higher tier only. A fractional index is a root:

$a^{\frac{1}{n}} = \sqrt[n]{a}, \qquad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m} = \sqrt[n]{a^{m}}.$an1​=na​,anm​=(na​)m=nam​.

Take the root first (the denominator $n$n) to keep the numbers small, then raise to the power (the numerator $m$m).

Worked Example 6 [H]

Work out (a) $25^{\frac{1}{2}}$2521​, (b) $27^{\frac{1}{3}}$2731​, (c) $16^{\frac{3}{4}}$1643​.

(a) $25^{\frac{1}{2}} = \sqrt{25} = 5$2521​=25​=5. (b) $27^{\frac{1}{3}} = \sqrt[3]{27} = 3$2731​=327​=3. (c) $16^{\frac{3}{4}} = (\sqrt[4]{16})^{3} = 2^{3} = 8$1643​=(416​)3=23=8.

Answers: (a) $5$5, (b) $3$3, (c) $8$8.

Worked Example 7 [H]

Work out $8^{-\frac{2}{3}}$8−32​.

Deal with each part: the negative means reciprocal, the $\frac{1}{3}$31​ is a cube root, the $2$2 is a square.

$8^{-\frac{2}{3}} = \dfrac{1}{8^{\frac{2}{3}}} = \dfrac{1}{(\sqrt[3]{8})^{2}} = \dfrac{1}{2^{2}} = \dfrac{1}{4}.$8−32​=832​1​=(38​)21​=221​=41​.

Answer: $\dfrac{1}{4}$41​.

Common error: Cubing instead of cube-rooting, or forgetting the reciprocal from the minus sign. Handle the sign, the root, and the power as three separate steps.

Powers of Negative Numbers

When the base is negative, the bracket matters. A negative base raised to an even power gives a positive result; raised to an odd power it stays negative, because the minus signs pair up.

Worked Example 7b

Work out (a) $(-3)^{2}$(−3)2, (b) $(-3)^{3}$(−3)3, (c) $-3^{2}$−32.

(a) $(-3)^{2} = (-3) \times (-3) = 9$(−3)2=(−3)×(−3)=9 (two negatives multiply to a positive). (b) $(-3)^{3} = (-3) \times (-3) \times (-3) = -27$(−3)3=(−3)×(−3)×(−3)=−27 (three negatives give a negative). (c) $-3^{2}$−32 has no bracket, so only the $3$3 is squared and the minus stays outside: $-3^{2} = -(3 \times 3) = -9$−32=−(3×3)=−9.

Answers: (a) $9$9, (b) $-27$−27, (c) $-9$−9.

Common error: Reading $-3^{2}$−32 as $(-3)^{2}$(−3)2. Without a bracket the power binds to the $3$3 alone, so $-3^{2} = -9$−32=−9, not $+9$+9. This single distinction is a classic Paper 1 discriminator.

OCR Exam Tip: In general, a negative base to an even power is positive and to an odd power is negative — count the factors. Always check whether the question has brackets around the negative.

Extended Worked Examples

Worked Example 8: Combining several laws

Simplify $\dfrac{(2a^{3})^{2} \times a^{-1}}{a^{4}}$a4(2a3)2×a−1​.

First $(2a^{3})^{2} = 4a^{6}$(2a3)2=4a6. Numerator: $4a^{6} \times a^{-1} = 4a^{5}$4a6×a−1=4a5. Divide: $\dfrac{4a^{5}}{a^{4}} = 4a^{1} = 4a$a44a5​=4a1=4a.

Answer: $4a$4a.

Worked Example 9: A power of a power with a coefficient

Simplify $(3x^{2}y)^{3}$(3x2y)3.

Raise every factor to the power $3$3: $3^{3} \times x^{2 \times 3} \times y^{3} = 27x^{6}y^{3}$33×x2×3×y3=27x6y3.

Answer: $27x^{6}y^{3}$27x6y3.

Common error: Forgetting to cube the $3$3 as well, giving $3x^{6}y^{3}$3x6y3.

Worked Example 10 [H]: Negative fractional index of a fraction

Work out $\left(\dfrac{16}{81}\right)^{-\frac{3}{4}}$(8116​)−43​.

The negative index flips the fraction: $\left(\dfrac{81}{16}\right)^{\frac{3}{4}}$(1681​)43​. Take the fourth root of top and bottom, then cube: $\dfrac{(\sqrt[4]{81})^{3}}{(\sqrt[4]{16})^{3}} = \dfrac{3^{3}}{2^{3}} = \dfrac{27}{8}$(416​)3(481​)3​=2333​=827​.

Answer: $\dfrac{27}{8}$827​.

Worked Example 11: Solving with indices

Work out the value of $n$n if $2^{n} = 32$2n=32.

Write $32$32 as a power of $2$2: $32 = 2^{5}$32=25. So $n = 5$n=5.