Recall the multiplication law: a^m × a^n = a^m+n Add the indices: x^5+3 Work out the sum: x^8 Answer: x^8

Recall the division law: a^ma^n = a^m-n Subtract the indices: y^9-4 Work out the difference: y^5 Answer: y^5

Recall the power law: (a^m)^n = a^mn Multiply the indices: a^4 × 3 Work out the product: a^12 Answer: a^12

Recall the zero-index rule: a^0 = 1 (a ≠ 0) See why it works: 5^25^2 = 5^2-2 = 5^0 Apply the rule: 5^0 = 1 Answer: 1

Recall the negative-index rule: a^-n = 1a^n Rewrite with a positive index: 3^-2 = 13^2 Work out the power: (1)/(9) Answer: (1)/(9)

Indices and surds Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Indices and surds questions. See exactly how to solve problems on index laws, multiplication of powers, division of powers, power of a power.

index lawsmultiplication of powersdivision of powerspower of a powerzero indexnegative indices

A-Level70 questionsStep-by-step solutions

Question 1

1 markeasy

Simplify

x^{5} \times x^{3}

Worked solution

Recall the multiplication law
$a^{m} \times a^{n} = a^{m+n}$
When you multiply powers that share the same base you keep the base and add the indices. Here the base is $x$ in both parts. This is one of the basic laws of indices.
Add the indices
$x^{5+3}$
The two indices are 5 and 3, so we add them. The base $x$ does not change.
Work out the sum
$x^{8}$
Since $5+3=8$ , the answer is $x$ to the power 8. That is already in its simplest form.

Answer

x^{8}

Question 2

1 markeasy

Simplify

\dfrac{y^{9}}{y^{4}}

Worked solution

Recall the division law
$\frac{a^{m}}{a^{n}} = a^{m-n}$
When you divide powers with the same base you keep the base and subtract the indices. The base here is $y$ . Remember to subtract the bottom index from the top one.
Subtract the indices
$y^{9-4}$
The top index is 9 and the bottom index is 4, so we compute $9-4$ .
Work out the difference
$y^{5}$
Because $9-4=5$ , the simplified answer is $y^{5}$ .

Answer

y^{5}

Question 3

1 markeasy

Simplify

\left(a^{4}\right)^{3}

Worked solution

Recall the power law
$\left(a^{m}\right)^{n} = a^{mn}$
When a power is raised to another power you keep the base and multiply the indices. Do not add them here. This is the third index law.
Multiply the indices
$a^{4 \times 3}$
The indices 4 and 3 are multiplied together.
Work out the product
$a^{12}$
Since $4 imes3=12$ , the answer is $a^{12}$ .

Answer

a^{12}

Question 4

1 markeasy

Evaluate

5^{0}

Worked solution

Recall the zero-index rule
$a^{0} = 1 \quad (a \neq 0)$
Any non-zero number raised to the power 0 is equal to 1. This looks surprising but follows from the division law. It is worth memorising.
See why it works
$\frac{5^{2}}{5^{2}} = 5^{2-2} = 5^{0}$
Dividing any power by itself gives 1, but the division law also gives $5^0$ . So $5^0$ must equal 1.
Apply the rule
$5^{0} = 1$
Here the base is 5, which is not zero, so the value is simply 1.

Answer

1

Question 5

1 markeasy

Evaluate

3^{-2}

Worked solution

Recall the negative-index rule
$a^{-n} = \frac{1}{a^{n}}$
A negative index means take the reciprocal (one over) the positive power. The sign of the index tells you to flip, not to make the number negative.
Rewrite with a positive index
$3^{-2} = \frac{1}{3^{2}}$
We move the power to the denominator and change the index to $+2$ .
Work out the power
$\frac{1}{9}$
Since $3^{2}=9$ , the value is $\tfrac{1}{9}$ .

Answer

\frac{1}{9}

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