To solve for [tex]\(\sqrt{343}\)[/tex] in its simplest form, let’s follow a systematic approach:
1. Prime Factorization of 343:
[tex]\(343\)[/tex] can be broken down with its prime factors. Upon examining it, we find:
[tex]\[
343 = 7 \times 7 \times 7 = 7^3
\][/tex]
2. Taking the Square Root:
To find the square root of [tex]\(343\)[/tex], we use the property of square roots:
[tex]\[
\sqrt{343} = \sqrt{7^3} = \sqrt{7^2 \times 7}
\][/tex]
3. Simplifying the Expression:
By properties of square roots, [tex]\(\sqrt{7^2 \times 7}\)[/tex] can be simplified as:
[tex]\[
\sqrt{7^2 \times 7} = 7 \sqrt{7}
\][/tex]
So, the simplest form of [tex]\(\sqrt{343}\)[/tex] is:
[tex]\[
7 \sqrt{7}
\][/tex]
Thus, the correct answer is:
[tex]\(7 \sqrt{7}\)[/tex]
