Question

a. \\(\sqrt{324}\\) b. \\(\sqrt{864}\\)

Explanation:

Part a

Step1: Identify perfect square

We know that \( 18\times18 = 324 \), so \( \sqrt{324} \) is the square root of a perfect square.
\( \sqrt{324}=\sqrt{18\times18} \)

Step2: Simplify the square root

Using the property \( \sqrt{a\times a}=a \) for non - negative \( a \), we get \( \sqrt{18\times18} = 18 \)

Step1: Factor the number inside the square root

First, we factor \( 864 \) into prime factors. We know that \( 864=144\times6 \), and \( 144 = 12\times12=12^{2} \)
So \( \sqrt{864}=\sqrt{144\times6} \)

Step2: Use the square root property

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (where \( a\geq0,b\geq0 \)), we have \( \sqrt{144\times6}=\sqrt{144}\times\sqrt{6} \)
Since \( \sqrt{144} = 12 \), then \( \sqrt{144}\times\sqrt{6}=12\sqrt{6} \)

Answer:

\( 18 \)

Part b