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Numeracy · Topic 4 of 6

Indices

Video lesson7 worked examples

Theory

Candidates must be able to multiply and divide using positive and negative indices, including fractions.

Key laws of indices to apply include:

(ab)^m = a^m b^m

(a^m)^n = a^{mn}

a^{m/n} = \sqrt[n]{a^m}

x^{-a} = \frac{1}{x^a}

Where possible, applying the laws in combination is desirable as preparation for Higher Maths.

The Golden Rule: the laws only apply when the bases match. With the same base, add the indices to multiply and subtract them to divide. A negative index means “one over”, and a fractional index means a root.

⚠️ Common Examiner Traps

Ignoring the coefficient: $(2a)^3 = 8a^3$ , not $2a^3$ — the $2$ is cubed as well.
A negative index is not a negative number: $x^{-2} = \frac{1}{x^2}$ , which is positive. The sign of the index controls the reciprocal, not the sign of the answer.
Root-and-power order: in $a^{m/n}$ the denominator $n$ is the root and the numerator $m$ is the power — take the root first to keep the numbers small.
Forgetting $a^0 = 1$ : anything (except 0) to the power zero is 1, not 0.

Worked examples

Example 1

Multiplication with Indices

Simplify $2a^3 \times 4a^{-1}$ .

Step 1: Multiply the numbers: 2 × 4 = 8.

Step 2: Add the powers (since the bases are the same): 3 + (−1) = 2.

Answer: $8a^2$ .

Example 2

Multiplication and Division Together

Simplify $\dfrac{m^6 \times m^2}{m^3}$ .

Step 1: Deal with the top first — multiplying means adding the powers: $m^6 \times m^2 = m^{6+2} = m^8$ .

Step 2: Now divide, which means subtracting the power on the bottom: $m^{8} \div m^3 = m^{8-3}$ .

Answer: $m^5$ .

Example 3

Raising a Power to a Power

Simplify $(3p^4)^2$ .

Step 1: The power outside applies to everything inside the bracket, including the 3. Deal with each part: $3^2 = 9$ .

Step 2: For the letter, multiply the powers: $(p^4)^2 = p^{4 \times 2} = p^8$ .

Answer: $9p^8$ .

Example 4

Negative Indices

Simplify $(2t^{-3})^2$ , giving your answer with a positive index.

Step 1: Apply the outside power to each part: $2^2 = 4$ and $(t^{-3})^2 = t^{-6}$ , giving $4t^{-6}$ .

Step 2: A negative index means “one over”, so move the $t$ to the denominator to make the index positive:

4t^{-6} = \frac{4}{t^6}

Answer: $\dfrac{4}{t^6}$ .

Example 5

Fractional Indices

Evaluate $8^{2/3}$ .

Step 1: Apply the rule $a^{m/n} = \sqrt[n]{a^m}$ . The denominator (3) becomes the root, and the numerator (2) becomes the power: $(\sqrt[3]{8})^2$ .

Step 2: The cube root of 8 is 2. Then square it: $2^2 = 4$ .

Answer: 4.

Example 6

Writing in the Form $x^n$

Express $\dfrac{1}{\sqrt{x}}$ in the form $x^n$ .

Step 1: A square root is a power of $\tfrac{1}{2}$ : $\sqrt{x} = x^{1/2}$ .

Step 2: The whole thing is one over that, and “one over” makes the index negative: $\dfrac{1}{x^{1/2}} = x^{-1/2}$ .

Answer: $x^{-1/2}$ .

Example 7

🔗 Bringing it together

Simplify $\dfrac{(2x^3)^4}{4x^5}$ .

Step 1: Deal with the bracket on top first, applying the power to a power to both the 2 and the $x$ : $(2x^3)^4 = 2^4 x^{12} = 16x^{12}$ .

Step 2: The expression is now $\dfrac{16x^{12}}{4x^5}$ . Divide the numbers and subtract the powers: $16 \div 4 = 4$ and $x^{12-5} = x^7$ .

Answer: $4x^7$ .

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13 Indices questions with answers and video solutions