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Finding Square Roots

Find square roots using prime factorisation.

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Defining Square Roots

A square root undoes squaring.

If the area of a square is 49 sq. cm, the length of its side is 7 cm because $7 \times 7 = 49$ , or $7^2 = 49$ .

In mathematics, we say that 7 is the square root of 49. In general, if $y = x^2$ , then $x$ is the square root of $y$ .

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Square Root Symbol

The square root symbol, and how to write it.

Formula

n^{2} = n

Inverse operation of a square

Example

64 = 8

Since 8 × 8 = 64

Condition

Positive Roots Only

We only consider the positive square root denoted by the symbol

Roots via Prime Factorisation

Worked example: the square root of 324.

Problem. Is 324 a perfect square? If so, find its square root.

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Identifying Non-Squares via Factors

Worked example: is 156 a perfect square?

Problem. Is 156 a perfect square?

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Find Root of 1156

Your turn: find the square root of 1156.

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To find the square root of $1156$ using prime factorization, we start by finding its prime factors: $2 \times 2 \times 17 \times 17$ . Next, we attempt to divide these prime factors into two identical groups. We can arrange them as $(2 \times$ $) \times (2 \times 17)$ . Since we can perfectly split the factors into two equal groups, the product of the factors in either group combines to form the square root. Evaluating this product, we calculate $2 \times 17 = 34$ . Therefore, we can confidently state that $\sqrt{1156}$ is .

Spot the Correct Root

Choose the correct square root.

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A number factors as $2^4 \times 3^2$ . What is its square root?

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