To simplify the expression [tex]\(\frac{1}{2} \sqrt{108 a^5 b^7}\)[/tex], let's break it down step by step.
1. Simplify the square root of the numerical part:
[tex]\[
\sqrt{108}
\][/tex]
We factorize 108 to its prime factors:
[tex]\[
108 = 2^2 \times 3^3
\][/tex]
Therefore,
[tex]\[
\sqrt{108} = \sqrt{2^2 \times 3^3} = \sqrt{2^2} \times \sqrt{3^3} = 2 \times 3 \sqrt{3} = 6 \sqrt{3}
\][/tex]
2. Simplify the square root of the variable [tex]\(a\)[/tex]:
[tex]\[
\sqrt{a^5}
\][/tex]
We can split [tex]\(a^5\)[/tex] into:
[tex]\[
a^5 = a^4 \times a = (a^2)^2 \times a
\][/tex]
Therefore,
[tex]\[
\sqrt{a^5} = \sqrt{(a^2)^2 \times a} = \sqrt{(a^2)^2} \times \sqrt{a} = a^2 \sqrt{a}
\][/tex]
3. Simplify the square root of the variable [tex]\(b\)[/tex]:
[tex]\[
\sqrt{b^7}
\][/tex]
We can split [tex]\(b^7\)[/tex] into:
[tex]\[
b^7 = b^6 \times b = (b^3)^2 \times b
\][/tex]
Therefore,
[tex]\[
\sqrt{b^7} = \sqrt{(b^3)^2 \times b} = \sqrt{(b^3)^2} \times \sqrt{b} = b^3 \sqrt{b}
\][/tex]
4. Combine all the simplified parts:
[tex]\[
\frac{1}{2} \sqrt{108 a^5 b^7} = \frac{1}{2} \times 6 \sqrt{3} \times a^2 \sqrt{a} \times b^3 \sqrt{b}
\][/tex]
Let's combine the coefficients and the square roots:
[tex]\[
\frac{1}{2} \times 6 \times a^2 \times b^3 \times \sqrt{3 \times a \times b} = 3 a^2 b^3 \sqrt{3ab}
\][/tex]
So, the simplified form of [tex]\(\frac{1}{2} \sqrt{108 a^5 b^7}\)[/tex] is:
[tex]\[
3 a^2 b^3 \sqrt{3ab}
\][/tex]
