To simplify the radical expression [tex]\(\sqrt{48 x^4 y^3 z^9}\)[/tex], follow these steps:
1. Factorize the constants and the variables:
[tex]\[
\sqrt{48 x^4 y^3 z^9}
\][/tex]
We start by breaking down each component inside the square root:
- [tex]\(48\)[/tex] can be factored as [tex]\(16 \times 3\)[/tex], which is [tex]\(4^2 \times 3\)[/tex].
- [tex]\(x^4\)[/tex] is a perfect square since [tex]\(x^4 = (x^2)^2\)[/tex].
- [tex]\(y^3\)[/tex] can be split as [tex]\(y^2 \times y\)[/tex].
- [tex]\(z^9\)[/tex] can be written as [tex]\((z^4)^2 \times z\)[/tex].
2. Rewrite the expression using these factors:
[tex]\[
\sqrt{48 x^4 y^3 z^9} = \sqrt{(4^2 \times 3) x^4 (y^2 y) (z^4 z^4 z)}
\][/tex]
3. Simplify by taking square roots of the perfect squares:
- Take the square root of [tex]\(4^2\)[/tex], which is [tex]\(4\)[/tex].
- Take the square root of [tex]\(x^4\)[/tex], which is [tex]\(x^2\)[/tex].
- Take the square root of [tex]\(y^2\)[/tex], which is [tex]\(y\)[/tex].
- Take the square root of [tex]\((z^4)^2\)[/tex], which is [tex]\(z^4\)[/tex].
Thus, the expression can be simplified to:
[tex]\[
4 x^2 y z^4 \sqrt{3 y z}
\][/tex]
4. Combine all the simplified terms:
Since [tex]\( \sqrt{3 y z}\)[/tex] cannot be further simplified, the final simplified form of the radical expression is:
[tex]\[
4 x^2 y z^4 \sqrt{3 y z}
\][/tex]
However, taking it a step further, we note that the square roots can be further adjusted, leading to:
[tex]\[
4 \sqrt{3} x^2 y z^4 \sqrt{y z^1}
\][/tex]
So the completely simplified form of the given radical expression is:
[tex]\[
4 \sqrt{3} \sqrt{x^4 y^3 z^9}
\][/tex]
Therefore, the fully simplified radical expression is:
[tex]\[
4 \sqrt{3} \sqrt{x^4 y^3 z^9}
\][/tex]
