To express [tex]\(\sqrt{48}\)[/tex] in the form [tex]\(k \sqrt{3}\)[/tex], where [tex]\(k\)[/tex] is an integer, let's follow these steps:
1. Prime Factorization of 48:
[tex]\[
48 = 2 \times 24 \\
24 = 2 \times 12 \\
12 = 2 \times 6 \\
6 = 2 \times 3
\][/tex]
Therefore, the prime factorization of 48 is:
[tex]\[
48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3
\][/tex]
2. Rewrite [tex]\(\sqrt{48}\)[/tex] using its prime factors:
Substitute the prime factorization back into the square root:
[tex]\[
\sqrt{48} = \sqrt{2^4 \times 3}
\][/tex]
3. Simplify the square root:
Use the property of square roots that [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{48} = \sqrt{2^4 \times 3} = \sqrt{2^4} \times \sqrt{3}
\][/tex]
Since [tex]\(\sqrt{2^4} = \sqrt{16} = 4\)[/tex], we have:
[tex]\[
\sqrt{48} = 4 \times \sqrt{3}
\][/tex]
Hence, [tex]\(\sqrt{48}\)[/tex] can be written in the form [tex]\(k \sqrt{3}\)[/tex], where [tex]\(k = 4\)[/tex].
So, the integer [tex]\(k\)[/tex] is
[tex]\[
\boxed{4}
\][/tex]
