Squares, roots & exponents

Start here: powers are a shorthand

Writing $5 \times 5 \times 5 \times 5$ gets long. So we write $5^4$ — read as “five to the fourth” — meaning “five multiplied by itself four times”.

Squaring is the most common case. $5 \times 5$ is written $5^2$ — “five squared”. The word “squared” comes from the fact that a $5 \times 5$ square has area $25$ :

3 \text{ squared} = 3^2 = 9, \qquad 4 \text{ squared} = 4^2 = 16, \qquad 7 \text{ squared} = 7^2 = 49.

A square root reverses the squaring. $\sqrt{49}$ asks “what number squared gives $49$ ?” Answer: $7$ .

Year 7 core

By the end of this topic you should be able to:

recognise square numbers and take the square root of a perfect square,
solve problems involving squares and square roots,
write a natural number as a product of prime factors using exponent notation,
use powers of $10$ to write numbers in expanded notation.

Worked example 0 Real-world example: tiling a square patio

You buy $36$ square tiles, each $1$ m $\times$ $1$ m. You want to lay them in a square. How many tiles fit along each side?

The area is $36$ tiles (a square number).
Side length $= \sqrt{36} = 6$ .
You lay $6$ tiles along each side: $6 \times 6 = 36$ . ✓

Key idea: the square root “undoes” squaring — it takes you from the area back to the side length.

1. Index notation

Index form is shorthand for repeated multiplication. The number being multiplied is the base; the small raised number is the index (or exponent, or power).

5 \times 5 \times 5 \times 5 \;=\; 5^{4}.

Here $5$ is the base and $4$ is the index. We read $5^4$ as “five to the fourth” or “five to the power of four”.

2. Square numbers

A square number is the result of multiplying a whole number by itself:

1^2 = 1, \; 2^2 = 4, \; 3^2 = 9, \; 4^2 = 16, \; 5^2 = 25, \; 6^2 = 36, \; 7^2 = 49, \; 8^2 = 64, \; 9^2 = 81, \; 10^2 = 100.

Square numbers take their name from a visual pattern: $3^2 = 9$ is the number of dots in a $3 \times 3$ square.

3. Square roots of perfect squares

The square root asks “what number, multiplied by itself, gives this value?”

Square root

\sqrt{a} \;=\; b \quad \text{means} \quad b \geq 0 \text{ and } b^{2} = a.

$\sqrt{49} = 7$ because $7^2 = 49$ .

4. Powers of 10 and expanded notation

Powers of $10$ match the place-value columns:

10^1 = 10, \; 10^2 = 100, \; 10^3 = 1000, \; 10^4 = 10\,000.

Any natural number can be written in expanded notation using these powers:

4528 \;=\; 4 \times 10^3 + 5 \times 10^2 + 2 \times 10^1 + 8.

5. Prime factorisation using exponents

Every natural number greater than $1$ can be written as a product of prime factors. When a prime appears more than once, use exponent notation for brevity.

Worked example E Very easy: prime factorise 12

What primes (building-block numbers) multiply to give $12$ ?

12 = 2 \times 6 = 2 \times 2 \times 3.

Two $2$ s and one $3$ . In exponent form: $12 = 2^2 \times 3$ .

Worked example 1 Prime factorise 72

Break $72$ down into primes using a factor tree:

Factor tree for 72. At each step, split into two factors until every leaf is prime (circled). Reading the leaves: 72 = 2 × 2 × 2 × 3 × 3.

In exponent form:

72 = 2^3 \times 3^2.

Worked example 2 Squaring a two-digit number using place value

Calculate $24^2$ .

24^2 = (20 + 4)^2 = 20 \times 20 + 2 \times 20 \times 4 + 4 \times 4 = 400 + 160 + 16 = 576.

(This uses an area-diagram approach rather than a calculator.)

Practice: Year 7 core

Fluency

Squares and square roots

Evaluate $3^2$ .
Evaluate $6^2$ .
Evaluate $12^2$ .
Evaluate $\sqrt{81}$ .
Evaluate $\sqrt{144}$ .
Evaluate $\sqrt{400}$ .
Between which two consecutive whole numbers does $\sqrt{50}$ lie?
Between which two consecutive whole numbers does $\sqrt{90}$ lie?
Which is bigger: $\sqrt{64}$ or $5^2$ ?
Evaluate $\sqrt{25} + \sqrt{36}$ .

Fluency

Powers of 10 and expanded notation

Evaluate $10^3$ .
Evaluate $10^5$ .
Write $3427$ in expanded notation using powers of $10$ .
Write $50\,608$ in expanded notation using powers of $10$ .
Write the number that equals $7 \times 10^3 + 2 \times 10^2 + 6$ .

Fluency

Prime factorisation

Write $20$ as a product of primes in exponent form.
Write $36$ as a product of primes in exponent form.
Write $84$ as a product of primes in exponent form.
Write $100$ as a product of primes in exponent form.
Write $200$ as a product of primes in exponent form.

Reasoning

Explain and apply

Explain in your own words why $\sqrt{9 + 16} \neq \sqrt{9} + \sqrt{16}$ .
Without a calculator, decide whether $\sqrt{200}$ is closer to $14$ or $15$ . Justify.
A square garden bed has an area of $144$ m^2. What is the length of one side?
Find the highest common factor of $72$ and $120$ by comparing their prime factorisations.
Find the lowest common multiple of $12$ and $18$ by using prime factorisation.

Extension - index laws

Index laws (positive whole-number indices)

Multiplying: add the indices

a^m \times a^n \;=\; a^{m+n}. \qquad \text{Example: } 2^3 \times 2^5 = 2^8.

Dividing: subtract the indices

a^m \div a^n \;=\; a^{m-n}, \quad a \neq 0. \qquad \text{Example: } 5^7 \div 5^3 = 5^4.

Power of a power: multiply the indices

(a^m)^n \;=\; a^{m \times n}. \qquad \text{Example: } (3^2)^4 = 3^8.

Practice: Extension

Fluency

Using the index laws

Simplify $2^4 \times 2^5$ .
Simplify $a^6 \times a^2$ .
Simplify $5^{10} \div 5^4$ .
Simplify $x^7 \div x^3$ .
Simplify $(3^2)^3$ .
Simplify $(m^4)^5$ .
Simplify $\dfrac{y^{10}}{y^4}$ .
Simplify $\dfrac{m^6 \times m^3}{m^5}$ .

Reasoning

Spot the mistake

Tim writes $2^3 + 2^3 = 2^6$ . Is Tim correct? If not, what has gone wrong?
Leah writes $(a^2)^3 = a^5$ . Explain Leah’s error and give the correct simplification.
Simplify $\dfrac{p^2 q^3 \times p^4 q}{p^3 q^2}$ .
A bacterium doubles every hour. Starting from one cell, how many cells are there after $6$ hours? Write the answer as a power of $2$ .