Squares and Square Roots

Squaring a number, undoing it with a square root, and finding the square root of a perfect square from its prime factorisation

Before you start

You can already write a number as a product of its prime factors in index notation:

$72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^{3} \times 3^{2}$

The index shows how many times a prime factor is multiplied. A prime factor written with no index has index 1: $2^{2} \times 7$ means $2^{2} \times 7^{1}$ .

Squares and Perfect Squares

To square a number, multiply it by itself. In index notation, $5 \times 5 = 5^{2} = 25$ . We read $5^{2}$ as "5 squared", and we say that 25 is the square of 5.

A square of side 5 cm is made of 5 rows of 5 one-centimetre squares, so its area is $5 \times 5 = 25 \text{ cm}^{2}$ . That is why $5^{2}$ is read as "5 squared".

Definition:

The square of a number is the number multiplied by itself: the square of $a$ is $a \times a = a^{2}$ . A perfect square is the square of a whole number.

The squares of 1 to 12 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144. They are worth learning by heart.

Square Roots

Finding a square root undoes squaring. Since $5^{2} = 25$ , the square root of 25 is 5. We write $\sqrt{25} = 5$ .

Definition:

If $a \times a = b$ and $a$ is positive (greater than 0), then $a$ is the positive square root of $b$ . We write $\sqrt{b} = a$ . The sign $\sqrt{\phantom{9}}$ is the square root sign, and it always stands for the positive square root.

25 has a second square root, the negative number $-5$ . You will meet it when you learn to multiply negative numbers. $\sqrt{25}$ means 5 only.

Examples

$6^{2} = 36$ , so $\sqrt{36} = 6$

$9^{2} = 81$ , so $\sqrt{81} = 9$

$12^{2} = 144$ , so $\sqrt{144} = 12$

Perfect Squares in Index Notation

Squaring a number doubles every index in its prime factorisation. Take $18 = 2 \times 3^{2}$ :

$18^{2} = (2 \times 3^{2}) \times (2 \times 3^{2}) = 2^{2} \times 3^{4}$

Each prime factor is now there twice as many times, so the indices 1 and 2 have become 2 and 4. Doubling always gives an even number, and that gives a test for perfect squares.

Rule:

A whole number greater than 1 is a perfect square if every index in its prime factorisation is even. If any index is odd, it is not a perfect square.

Example 1: Which are perfect squares?

Which of these numbers are perfect squares?

(a) $2^{4} \times 5^{2}$

(b) $2^{3} \times 3^{2}$

Solution:

(a) The indices are 4 and 2. Both are even, so $2^{4} \times 5^{2}$ is a perfect square.

(b) The index of 2 is 3, which is odd, so $2^{3} \times 3^{2}$ is not a perfect square.

Answer: Only (a) is a perfect square. It is $400 = 20^{2}$ .

Finding a Square Root by Prime Factorisation

When every index is even, the prime factors can be shared into two equal groups. Each group gets half of every index, and one group is the square root.

Method: Square Root by Prime Factorisation

Write the number as a product of its prime factors in index notation.
Check that every index is even.
Halve each index.
Multiply to get the square root.

Example 2: Find $\sqrt{324}$

Step 1: Divide 324 by prime numbers until you reach 1, then write it as a product of its prime factors.

$324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 = 2^{2} \times 3^{4}$

Step 2: Both indices are even. Share the prime factors into two equal groups.

$324 = (2 \times 3^{2}) \times (2 \times 3^{2}) = (2 \times 3^{2})^{2}$

Step 3: One group is the square root. Its indices, 1 and 2, are half of 2 and 4.

$\sqrt{324} = 2 \times 3^{2} = 2 \times 9 = 18$

Answer: $\sqrt{324} = 18$

Check: $18 \times 18 = 324$

Example 3: Find $\sqrt{1764}$

Step 1: Write 1764 as a product of its prime factors.

$1764 = 2 \times 2 \times 3 \times 3 \times 7 \times 7 = 2^{2} \times 3^{2} \times 7^{2}$

Step 2: Every index is even, so 1764 is a perfect square. Halve each index.

$\sqrt{1764} = 2 \times 3 \times 7 = 42$

Answer: $\sqrt{1764} = 42$

Check: $42 \times 42 = 1764$

Making a Perfect Square

A number with an odd index is not a perfect square. To make it one, every odd index must become even: multiply by that prime factor once more, or divide by it once.

Example 4: Making a perfect square

(a) Find the smallest positive whole number that 504 must be multiplied by to give a perfect square.

(b) Find the smallest positive whole number that 504 must be divided by to give a perfect square.

Step 1: Write 504 as a product of its prime factors.

$504 = 2 \times 2 \times 2 \times 3 \times 3 \times 7 = 2^{3} \times 3^{2} \times 7$

Step 2: Look for the odd indices. The indices of 2 and 7 are 3 and 1, which are odd. The index of 3 is already even.

(a) Multiply by one more 2 and one more 7. Then every index is even.

$504 \times (2 \times 7) = 2^{4} \times 3^{2} \times 7^{2}$

(b) Divide by one 2 and one 7. No 7 is left, and the indices that are left are even.

$504 \div (2 \times 7) = 2^{2} \times 3^{2}$

Answer: (a) The smallest number to multiply by is $2 \times 7 = 14$ . It gives $504 \times 14 = 7056 = 84^{2}$ .

(b) The smallest number to divide by is also $2 \times 7 = 14$ . It gives $504 \div 14 = 36 = 6^{2}$ .

Squares and Square Roots in Word Problems

The area of a square is the square of the length of its side. So the length of the side is the square root of the area.

Example 5: The side of a square

A square tabletop has an area of $5625 \text{ cm}^{2}$ . Find the length of its side.

Step 1: Length of side $= \sqrt{5625}$ cm. Write 5625 as a product of its prime factors.

$5625 = 3 \times 3 \times 5 \times 5 \times 5 \times 5 = 3^{2} \times 5^{4}$

Step 2: Both indices are even. Halve each index.

$\sqrt{5625} = 3 \times 5^{2} = 3 \times 25 = 75$

Answer: The length of its side is 75 cm.

Check: $75 \times 75 = 5625$

Common Mistakes

Doubling instead of squaring. $5^{2}$ means $5 \times 5 = 25$ , not $5 \times 2 = 10$ .
Halving the number instead of finding its square root. $\sqrt{36}$ is 6, not 18. Halve the indices of the prime factorisation, never the number itself.
Forgetting an index of 1. $3^{2} \times 7$ is not a perfect square, because the index of 7 is 1, which is odd.

Practice

Work each question on paper before you open its solution.

Question 1

Find the value of each of these.

(a) $9^{2}$

(b) $13^{2}$

(d) $\sqrt{121}$

ShowHide solution

Solution:

(a) $9^{2} = 9 \times 9 = 81$

(b) $13^{2} = 13 \times 13 = 169$

(d) $11 \times 11 = 121$ , so $\sqrt{121} = 11$

Answer: (a) 81 (b) 169 (c) 8 (d) 11

Question 2

Given that $45 = 3^{2} \times 5$ , write $45^{2}$ as a product of its prime factors in index notation.

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Solution:

$45^{2} = (3^{2} \times 5) \times (3^{2} \times 5)$

Each prime factor is there twice as many times, so each index is doubled: 2 becomes 4 and 1 becomes 2.

Answer: $45^{2} = 3^{4} \times 5^{2}$

Question 3

Which of these numbers are perfect squares? Find the square root of each one that is.

(a) $2^{6} \times 5^{2}$

(b) $2^{5} \times 7^{2}$

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Solution:

(a) The indices 6 and 2 are both even, so it is a perfect square. Halve each index: $\sqrt{2^{6} \times 5^{2}} = 2^{3} \times 5 = 8 \times 5 = 40$ .

(b) The index of 2 is 5, which is odd, so it is not a perfect square.

Answer: Only (a) is a perfect square. Its square root is 40.

Question 4

Find $\sqrt{784}$ by prime factorisation.

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Solution:

$784 = 2 \times 2 \times 2 \times 2 \times 7 \times 7 = 2^{4} \times 7^{2}$

Both indices are even. Halve each index: $\sqrt{784} = 2^{2} \times 7 = 4 \times 7 = 28$ .

Check: $28 \times 28 = 784$

Answer: $\sqrt{784} = 28$

Question 5

Find the smallest positive whole number that 180 must be multiplied by to give a perfect square. Then find the square root of that perfect square.

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Solution:

$180 = 2 \times 2 \times 3 \times 3 \times 5 = 2^{2} \times 3^{2} \times 5$

Only 5 has an odd index, 1. Multiply by one more 5: $180 \times 5 = 2^{2} \times 3^{2} \times 5^{2} = 900$ .

Halve each index: $\sqrt{900} = 2 \times 3 \times 5 = 30$ .

Answer: The smallest number to multiply by is 5. The perfect square is 900 and its square root is 30.

Question 6

Find the smallest positive whole number that 1350 must be divided by to give a perfect square.

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Solution:

$1350 = 2 \times 3 \times 3 \times 3 \times 5 \times 5 = 2 \times 3^{3} \times 5^{2}$

The indices of 2 and 3 are 1 and 3, which are odd. Divide by one 2 and one 3: $1350 \div (2 \times 3) = 3^{2} \times 5^{2}$ .

$1350 \div 6 = 225 = 15^{2}$

Answer: The smallest number to divide by is 6.

Question 7

The parade square of a school is a square with an area of $2304 \text{ m}^{2}$ . Find

(a) the length of its side,

(b) its perimeter.

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Solution:

(a) Length of side $= \sqrt{2304}$ m. Write 2304 as a product of its prime factors.

$2304 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^{8} \times 3^{2}$

Both indices are even. Halve each index: $\sqrt{2304} = 2^{4} \times 3 = 16 \times 3 = 48$ .

(b) A square has 4 equal sides. Perimeter $= 4 \times 48 = 192$ m.

Answer: (a) The length of its side is 48 m. (b) Its perimeter is 192 m.

Question 8

Mr Tan covers a square floor with 1296 square tiles, each of side 25 cm. The tiles fit exactly and none is cut.

(a) How many tiles are there along each side of the floor?

(b) Find the length of a side of the floor in metres.

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Solution:

(a) The floor is a square, so it has as many rows of tiles as there are tiles in each row. The number of tiles along each side, multiplied by itself, is 1296. So the number of tiles along each side is $\sqrt{1296}$ .

$1296 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 = 2^{4} \times 3^{4}$

Halve each index: $\sqrt{1296} = 2^{2} \times 3^{2} = 4 \times 9 = 36$ .

(b) Length of side $= 36 \times 25 = 900$ cm $= 9$ m.

Answer: (a) There are 36 tiles along each side. (b) The length of a side of the floor is 9 m.

Key Points

The square of a number is the number multiplied by itself: $a^{2} = a \times a$ .
A perfect square is the square of a whole number.
For a positive number $b$ , $\sqrt{b}$ is the positive number whose square is $b$ .
A whole number greater than 1 is a perfect square if every index in its prime factorisation is even, and is not one if any index is odd.
To square a number in index notation, double each index. To find the square root of a perfect square, halve each index.
To make a perfect square, multiply (or divide) by each prime factor that has an odd index, once.
The length of the side of a square is the square root of its area.

Squares and Square Roots

Before you start

Squares and Perfect Squares

Definition:

Square Roots

Definition:

Examples

Perfect Squares in Index Notation

Rule:

Example 1: Which are perfect squares?

Finding a Square Root by Prime Factorisation

Method: Square Root by Prime Factorisation

Example 2: Find 324\sqrt{324}324​

Example 3: Find 1764\sqrt{1764}1764​

Making a Perfect Square

Example 4: Making a perfect square

Squares and Square Roots in Word Problems

Example 5: The side of a square

Common Mistakes

Practice

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Key Points

Example 2: Find $\sqrt{324}$

Example 3: Find $\sqrt{1764}$