To solve the expression [tex]\(\sqrt{27 x^{12} y^8}\)[/tex], we'll break it down step by step and simplify each component separately before combining them.
1. Simplify the Square Root of the Constant Term:
[tex]\[
\sqrt{27}
\][/tex]
We start by factoring 27 into its prime factors:
[tex]\[
27 = 3^3
\][/tex]
So,
[tex]\[
\sqrt{27} = \sqrt{3^3} = \sqrt{3^2 \cdot 3} = 3 \sqrt{3}
\][/tex]
2. Simplify the Square Root of the Variable Term [tex]\(x^{12}\)[/tex]:
[tex]\[
\sqrt{x^{12}}
\][/tex]
Using the property of exponents for square roots [tex]\(\sqrt{x^n} = x^{n/2}\)[/tex], we get:
[tex]\[
\sqrt{x^{12}} = x^{12/2} = x^6
\][/tex]
3. Simplify the Square Root of the Variable Term [tex]\(y^8\)[/tex]:
[tex]\[
\sqrt{y^8}
\][/tex]
Similarly, applying the property of exponents:
[tex]\[
\sqrt{y^8} = y^{8/2} = y^4
\][/tex]
Now let's combine all the simplified parts:
Original expression:
[tex]\[
\sqrt{27 x^{12} y^8}
\][/tex]
Substituting the simplified parts:
[tex]\[
= \sqrt{27} \cdot \sqrt{x^{12}} \cdot \sqrt{y^8}
\][/tex]
[tex]\[
= 3 \sqrt{3} \cdot x^6 \cdot y^4
\][/tex]
So, the simplified expression is:
[tex]\[
3 \sqrt{3} x^6 y^4
\][/tex]
Therefore,
[tex]\[
\sqrt{27 x^{12} y^8} = 3 \sqrt{3} x^6 y^4
\][/tex]
