1.7 Perfect 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

\text{every index even} \iff \text{perfect square}, \qquad \sqrt{a^p \times b^q} = a^{p/2} \times b^{q/2}

Finding a square root using prime factorisation

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

All indices are even, so $1764$ is a perfect square.
$\sqrt{1764} = 2 \times 3 \times 7 \quad \scriptsize\textit{halve each index}$
$= 42$

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A practice question on this appears here. Practise primes hcf lcm.

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