Indices (Powers)
The Skill
What is an Index?
In $a^n$, the base is $a$ and the index (or power/exponent) is $n$.
$$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$$
Laws of Indices
Multiplying (same base) — ADD the powers
$$a^m \times a^n = a^{m+n}$$ Example: $3^2 \times 3^4 = 3^{2+4} = 3^6$
Dividing (same base) — SUBTRACT the powers
$$a^m \div a^n = a^{m-n}$$ Example: $5^7 \div 5^3 = 5^{7-3} = 5^4$
Power of a power — MULTIPLY the powers
$$(a^m)^n = a^{m \times n}$$ Example: $(2^3)^4 = 2^{3 \times 4} = 2^{12}$
Special Powers
Power of 0
$$a^0 = 1$$ (for any $a \neq 0$)
Power of 1
$$a^1 = a$$
Negative Indices (Higher)
$$a^{-n} = \frac{1}{a^n}$$ Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
Fractional Indices (Higher)
$$a^{\frac{1}{n}} = \sqrt[n]{a}$$ $$a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$$
Examples:
- $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$
- $27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$
The Traps
Common misconceptions and how to avoid them.
Applying index laws to different bases "The Base Blunder"
The Mistake in Action
Simplify $2^3 \times 3^2$
Wrong: $6^5$ or $2^5$
Why It Happens
Students try to apply the index laws even when the bases are different. The laws only work when the base is the same.
The Fix
Index laws only work when the bases are the same.
For $2^3 \times 3^2$, the bases are different (2 and 3), so you cannot combine them using index laws.
Just calculate: $$2^3 \times 3^2 = 8 \times 9 = 72$$
You can only use $a^m \times a^n = a^{m+n}$ when both terms have the same base $a$.
Spot the Mistake
Simplify $2^3 \times 3^2$
$= 6^5$
Click on the line that contains the error.
Multiplying base by fraction for fractional indices "The Fraction Fumble"
The Mistake in Action
Evaluate $16^{\frac{1}{2}}$
Wrong: $16 \times \frac{1}{2} = 8$
Why It Happens
Students don't recognise fractional indices as roots. They multiply the base by the fraction instead.
The Fix
A fractional index means a root:
$$a^{\frac{1}{n}} = \sqrt[n]{a}$$
So $a^{\frac{1}{2}} = \sqrt{a}$ (square root)
$$16^{\frac{1}{2}} = \sqrt{16} = 4$$
Memory aid: "Power of a half = square root" "Power of a third = cube root"
Spot the Mistake
Evaluate $16^{\frac{1}{2}}$
$= 16 \times \frac{1}{2} = 8$
Click on the line that contains the error.
Multiplying indices instead of adding when multiplying powers "The Power Multiplier"
The Mistake in Action
Simplify $a^3 \times a^4$
Wrong: $a^{3 \times 4} = a^{12}$
Why It Happens
Students confuse the rules for multiplying powers with the rule for a power of a power. They multiply the indices instead of adding them.
The Fix
When multiplying powers with the same base, ADD the indices.
$$a^m \times a^n = a^{m+n}$$
$$a^3 \times a^4 = a^{3+4} = a^7$$
Think about it: $a^3$ means $a \times a \times a$, and $a^4$ means $a \times a \times a \times a$. Together that's 7 $a$'s multiplied, so $a^7$.
Memory aid: "Times means Plus" for indices (when bases are the same)
Spot the Mistake
Simplify $a^3 \times a^4$
$= a^{3 \times 4} = a^{12}$
Click on the line that contains the error.
Thinking negative index gives negative answer "The Negative Confusion"
The Mistake in Action
Evaluate $4^{-2}$
Wrong: $4^{-2} = -16$
Why It Happens
Students confuse a negative index with a negative answer. They may think $4^{-2}$ means "negative $4^2$".
The Fix
A negative index means "one over" (reciprocal), NOT a negative answer.
$$a^{-n} = \frac{1}{a^n}$$
$$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$$
Note: $4^{-2}$ is a positive number (since we're dividing 1 by a positive number).
$-4^2$ (negative $4^2$) = $-16$ $4^{-2}$ (4 to the power of $-2$) = $\frac{1}{16}$
Spot the Mistake
Evaluate $4^{-2}$
$= -16$
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
Simplify $x^5 \times x^3$
Solution
When multiplying powers with the same base, add the indices.
$$x^5 \times x^3 = x^{5+3} = x^8$$
Answer: $x^8$
Check: $x^5$ means 5 $x$'s multiplied, $x^3$ means 3 $x$'s multiplied. Together: 8 $x$'s = $x^8$ ✓
Question
Simplify $\frac{y^9}{y^4}$
Solution
When dividing powers with the same base, subtract the indices.
$$\frac{y^9}{y^4} = y^{9-4} = y^5$$
Answer: $y^5$
Question
Simplify $(a^4)^3$
Solution
When raising a power to a power, multiply the indices.
$$(a^4)^3 = a^{4 \times 3} = a^{12}$$
Answer: $a^{12}$
Check: $(a^4)^3$ means $a^4 \times a^4 \times a^4 = a^{4+4+4} = a^{12}$ ✓
Question
Evaluate $27^{\frac{2}{3}}$
Solution
A fractional index $\frac{m}{n}$ means: take the $n$th root, then raise to the power $m$.
$$27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2$$
Step 1: Find the cube root of 27. $$\sqrt[3]{27} = 3$$ (since $3^3 = 27$)
Step 2: Square the result. $$3^2 = 9$$
Answer: 9
Alternative method: $27^{\frac{2}{3}} = (27^2)^{\frac{1}{3}} = \sqrt[3]{729} = 9$ (But finding cube root of 729 is harder!)
Level 2: Scaffolded
Fill in the key steps.
Question
Write $5^{-3}$ as a fraction.
Level 3: Solo
Try it yourself!
Question
Simplify $\frac{2x^5 \times 3x^2}{x^4}$
Show Solution
Step 1: Multiply the numerator. $$2x^5 \times 3x^2 = 6x^{5+2} = 6x^7$$
Step 2: Divide by $x^4$. $$\frac{6x^7}{x^4} = 6x^{7-4} = 6x^3$$
Answer: $6x^3$
Question
Solve $2^x = 32$
Show Solution
Write 32 as a power of 2: $$32 = 2^5$$
So the equation becomes: $$2^x = 2^5$$
Since the bases are equal, the indices must be equal: $$x = 5$$
Answer: $x = 5$
Check: $2^5 = 32$ ✓
Examiner's View
Mark allocation: Basic index laws: 1-2 marks. Negative/fractional indices: 2-3 marks.
Common errors examiners see:
- Multiplying indices when you should add
- Applying laws to different bases
- Confusing $a^{-n}$ with $-a^n$
- Errors with fractional indices
What gains marks:
- Showing each step of simplification
- Converting negative indices to fractions
- Evaluating numerical answers where possible
AQA Notes
AQA often combines indices with algebra — simplify $\frac{12x^5}{4x^2}$.