Determine the simplest product of the two square roots. sqrt18*sqrt6

Apply Product Rule: First, let's use the product rule for square roots, which says that sqrt(a) * sqrt(b) = sqrt(a * b). So, we have sqrt(18) * sqrt(6) = sqrt(18 * 6).Multiply Inside Square Root: Now, let's multiply the numbers inside the square root. 18 * 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 * 54, and then 54 can be broken down into 2 * 27, and 27 can be broken down into 3 * 9, and finally, 9 is 3 * 3. So, the prime factorization of 108 is 2 * 2 * 3 * 3 * 3.Pair Prime Factors: We can pair the prime factors inside the square root to simplify it. 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. So, sqrt(108) simplifies to 2 * 3 * sqrt(3), which is 6 * sqrt(3).Simplify Final Result: Therefore, the simplest form of the product of sqrt(18) and sqrt(6) is 6 * sqrt(3).

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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 $6 \times \sqrt{3}$ .