Full solution

Q. Determine the simplest product of the two square roots.

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\sqrt{18} \times \sqrt{6}

Apply Product Rule: First, let's use the product rule for square roots, which says that $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$ . $\newline$ So, we have $\sqrt{18} \times \sqrt{6} = \sqrt{18 \times 6}$ .
Multiply Inside Square Root: Now, let's multiply the numbers inside the square root. $18 \times 6$ equals $108$ , so we have $\sqrt{108}$ .
Find Prime Factors: Next, we need to simplify $\sqrt{108}$ . We can do this by finding the prime factors of $108$ . $108$ can be broken down into $2 \times 54$ , and then $54$ can be broken down into $2 \times 27$ , and $27$ can be broken down into $3 \times 9$ , and finally, $9$ is $3 \times 3$ . So, the prime factorization of $108$ is $108$ $1$ .
Pair Prime Factors: We can pair the prime factors inside the square root to simplify it. $\newline$ We have two pairs of $2$ 's and one pair of $3$ 's, which can be taken out of the square root as a single $2$ and a single $3$ . $\newline$ So, $\sqrt{108}$ simplifies to $2 \times 3 \times \sqrt{3}$ , which is $6 \times \sqrt{3}$ .
Simplify Final Result: Therefore, the simplest form of the product of $\sqrt{18}$ and $\sqrt{6}$ is $6 \times \sqrt{3}$ .