Let's tackle the problem step-by-step.
Step 1: Consider the given expression:
[tex]\[
\sqrt[3]{125 x} \cdot \sqrt[4]{256 x^2}
\][/tex]
Step 2: Rewrite the radicals as exponents. Recall that the cube root of a number can be written as raising the number to the power of [tex]\( \frac{1}{3} \)[/tex], and the fourth root can be written as raising the number to the power of [tex]\( \frac{1}{4} \)[/tex]. Therefore:
[tex]\[
\sqrt[3]{125 x} = (125 x)^{\frac{1}{3}}
\][/tex]
[tex]\[
\sqrt[4]{256 x^2} = (256 x^2)^{\frac{1}{4}}
\][/tex]
Step 3: Rewrite the expression by substituting these exponents back in:
[tex]\[
(125 x)^{\frac{1}{3}} \cdot (256 x^2)^{\frac{1}{4}}
\][/tex]
Step 4: Break each term inside the parentheses into their base components and apply the exponents:
[tex]\[
125 = 5^3 \implies (125 x)^{\frac{1}{3}} = (5^3 x)^{\frac{1}{3}} = 5^{3 \cdot \frac{1}{3}} x^{\frac{1}{3}} = 5 x^{\frac{1}{3}}
\][/tex]
[tex]\[
256 = 4^4 \implies (256 x^2)^{\frac{1}{4}} = (4^4 x^2)^{\frac{1}{4}} = 4^{4 \cdot \frac{1}{4}} (x^2)^{\frac{1}{4}} = 4 x^{\frac{2}{4}} = 4 x^{\frac{1}{2}}
\][/tex]
Step 5: Combine the simplified components:
[tex]\[
5 x^{\frac{1}{3}} \cdot 4 x^{\frac{1}{2}}
\][/tex]
Step 6: Multiply the numerical coefficients together:
[tex]\[
5 \cdot 4 = 20
\][/tex]
Step 7: Combine the exponents of [tex]\( x \)[/tex] by adding them together. We add the exponents because the bases are the same:
[tex]\[
x^{\frac{1}{3}} \cdot x^{\frac{1}{2}} = x^{\frac{1}{3} + \frac{1}{2}} = x^{\frac{2}{6} + \frac{3}{6}} = x^{\frac{5}{6}}
\][/tex]
Step 8: Put it all together:
[tex]\[
20 x^{\frac{5}{6}}
\][/tex]
We can express this result in simplest radical form:
[tex]\[
20 x^{\frac{5}{6}} = 20 x^{0.833333333333333}
\][/tex]
So, the expression [tex]\(\sqrt[3]{125 x} \cdot \sqrt[4]{256 x^2}\)[/tex] simplifies to:
[tex]\[
\boxed{20 x^{0.833333333333333}}
\][/tex]
This is our final simplified expression.
