Of course! Let's simplify the expression [tex]\(\sqrt{5 u w^5} \sqrt{15 u^8 w}\)[/tex] step by step, assuming all variables are positive real numbers.
1. Combine the expressions inside the square roots:
[tex]\[
\sqrt{5 u w^5} \sqrt{15 u^8 w} = \sqrt{(5 u w^5) (15 u^8 w)}
\][/tex]
2. Multiply the expressions under the square root:
[tex]\[
\sqrt{(5 u w^5) (15 u^8 w)} = \sqrt{5 \cdot 15 \cdot u \cdot u^8 \cdot w^5 \cdot w}
\][/tex]
3. Simplify the multiplication inside the square root:
[tex]\[
5 \cdot 15 = 75
\][/tex]
[tex]\[
u \cdot u^8 = u^9
\][/tex]
[tex]\[
w^5 \cdot w = w^6
\][/tex]
So the expression becomes:
[tex]\[
\sqrt{75 u^9 w^6}
\][/tex]
4. Break down the components of the square root:
[tex]\[
75 u^9 w^6
\][/tex]
5. Simplify each component separately inside the square root:
[tex]\[
\sqrt{75} = \sqrt{25 \cdot 3} = 5 \sqrt{3}
\][/tex]
[tex]\[
\sqrt{u^9} = u^{9/2} = u^4 \sqrt{u}
\][/tex]
[tex]\[
\sqrt{w^6} = w^{6/2} = w^3
\][/tex]
6. Combine the simplified parts:
[tex]\[
\sqrt{75 u^9 w^6} = 5 \sqrt{3} u^4 \sqrt{u} w^3
\][/tex]
7. Combine like terms:
[tex]\[
5 \sqrt{3} u^4 \sqrt{u} w^3 = 5 u^4 w^3 \sqrt{3u}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{5u^4w^3 \sqrt{3u}}
\][/tex]
