To simplify the expression [tex]\(\sqrt[3]{\frac{32 x^4}{54 x}}\)[/tex], we can follow these steps:
1. Simplify the fraction inside the cube root:
[tex]\[
\frac{32 x^4}{54 x}
\][/tex]
Here, we can simplify the numerical part and the variable part separately.
- Numerical part:
Simplify [tex]\(\frac{32}{54}\)[/tex]:
[tex]\[
\frac{32}{54} = \frac{32 \div 2}{54 \div 2} = \frac{16}{27}
\][/tex]
- Variable part:
Simplify [tex]\( \frac{x^4}{x} \)[/tex]:
[tex]\[
\frac{x^4}{x} = x^{4-1} = x^3
\][/tex]
So the fraction inside the cube root becomes:
[tex]\[
\frac{16}{27} x^3
\][/tex]
2. Apply the cube root to both the numerator and the variable separately:
Using the property that the cube root of a product is the product of the cube roots, we get:
[tex]\[
\sqrt[3]{\frac{16}{27} x^3} = \sqrt[3]{\frac{16}{27}} \cdot \sqrt[3]{x^3}
\][/tex]
- Cube root of the numerical part:
[tex]\[
\sqrt[3]{\frac{16}{27}} \approx 0.279982
\][/tex]
- Cube root of the variable part:
[tex]\[
\sqrt[3]{x^3} = x^{3 \cdot \frac{1}{3}} = x^1 = x
\][/tex]
Thus, we have:
[tex]\[
\sqrt[3]{\frac{16}{27} x^3} \approx 0.279982 \cdot x
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt[3]{\frac{32 x^4}{54 x}}\)[/tex] is approximately:
[tex]\[
0.279982 \cdot x
\][/tex]
