Squares and square roots

The square of a number is the number multiplied by itself. A square root undoes this: $49 = 7$ because $7 \times 7 = 49$ . Numbers like 1, 4, 9, 16 and 25 are perfect squares.

Formulas

Square and square root

n^{2} = n \times n n^{2} = n

Sum of the first n odd numbers

1 + 3 + 5 + \dots + (2 n - 1) = n^{2}

Square root by prime factorisation

pair up equal prime factors; take one from each pair

Worked examples

Example 1. Find

324

by prime factorisation.

$324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 = 2^{2} \times 3^{4}$ .
Take one factor from each pair: $2 \times 3 \times 3 = 18$ .

Answer: 18

Example 2. What is the smallest number by which 180 must be multiplied to get a perfect square?

$180 = 2^{2} \times 3^{2} \times 5$ . The factor 5 has no partner.
Multiply by 5: $900 = 3 0^{2}$ .

Answer: 5 (giving 900)

Example 3. Between which two whole numbers does

50

lie?

$7^{2} = 49$ and $8^{2} = 64$ .
Since $49 < 50 < 64, 7 < 50 < 8$ , and it is just above 7.

Answer: Between 7 and 8

Example 4. Is 2,000 a perfect square?

$4 4^{2} = 1936$ and $4 5^{2} = 2025$ , and 2000 lies between them.

Answer: No

Common mistakes

Writing $a + b = a + b$ . For example $9 + 16 = 5$ , not $3 + 4 = 7$ .
Forgetting that a perfect square cannot end in 2, 3, 7 or 8.
Taking one prime factor per pair incorrectly: you take one from each pair, not both.

Practice questions with answers

1. Find

81

A9
B8
C40.5
D18

Show answer

Answer: A) 9

9 \times 9 = 81. S o 81 = 9

2. What is the value of

81

Show answer

Answer: C) 9

9 \times 9 = 81, so 81 = 9

3. Which of these digits can a perfect square NEVER end with?

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Answer: B) 7

A perfect square can end only in 0, 1, 4, 5, 6 or 9. It can never end in 2, 3, 7 or 8. So 7 is impossible.

4. Which is larger,

0.49

or 0.49?

A0.49, because taking a square root always makes a number smaller
BThey are exactly equal
C $0.49$ , because it equals 0.07
D $0.49$ , because it equals 0.7 and 0.7 > 0.49

Show answer

Answer: D)

0.49

, because it equals 0.7 and 0.7 > 0.49

0.49 = 0.7

since

0.7 \times 0.7 = 0.49

, and 0.7 is larger than 0.49. A square root only shrinks a number that is greater than 1; for a number between 0 and 1 the root is LARGER than the number itself, which is why the first option fails. The option 0.07 misplaces the decimal point:

0.0 7^{2} = 0.0049

, not 0.49.

5. Find the value of

\frac{1296}{36}

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Answer: A) 6

1296 = 36

and

36 = 6

, so 36 / 6 = 6. (Equivalently

\frac{1296}{36} = 36 = 6.

)

6. Consider perfect squares whose unit digit is 4, such as 144 and 324.
Among the squares from

1^{2}

2 0^{2}

, how many end in the digit 4?

Show answer

Answer: B) 4

A square ends in 4 when the number ends in 2 or 8. From 1 to 20: 2, 8, 12, 18 — that is 4 numbers (4, 64, 144, 324).

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Frequently asked questions

What is a perfect square?

A number that is the square of a whole number: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on.

How do you find a square root by prime factorisation?

Split the number into prime factors, pair equal factors, and multiply one factor from each pair.

How can I tell quickly that a number is not a perfect square?

Check the last digit: perfect squares end only in 0, 1, 4, 5, 6 or 9.

What is the square root of a negative number?

There is no real square root, because no real number multiplied by itself gives a negative result.

Squares and square roots

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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