1.7Perfect Squares and Square Roots

A perfect square is a whole number whose square root is also whole: $0, 1, 4, 9, 16, 25, \ldots$ Its prime factorisation gives it away. In $900 = 2^2 \times 3^2 \times 5^2$ every index is even, so the factors split into two identical halves, and halving each index gives the root.

Perfect square test, and its root

\displaystyle \text{Perfect square exactly when every index is even}, \qquad \sqrt{a^p \times b^q} = a^{p/2} \times b^{q/2}

Every index pairs off exactly, so

900

is a perfect square. In

720

the

5

has nothing to pair with.

Finding a square root using prime factorisation

Given that $\displaystyle 1764 = 2^2 \times 3^2 \times 7^2$ , find $\displaystyle \sqrt{1764}$ .

Show solution▾

All indices are even, so

1764

is a perfect square.

\displaystyle \sqrt{1764}

\displaystyle = 2 \times 3 \times 7 \quad \scriptsize\textit{halve each index}

\displaystyle = 42

Now you try

Given that $900 = 2^{2} \times 3^{2} \times 5^{2}$ , find $\sqrt{900}$ . Give your answer as a product of its prime factors.

Show solution▾

\displaystyle \sqrt{900}

\displaystyle = \sqrt{2^{2} \times 3^{2} \times 5^{2}}

\displaystyle = 2 \times 3 \times 5 \quad \scriptsize\textit{halve each index}

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