To simplify the expression [tex]\(\sqrt[3]{128 x^{12} y^{13} z^{16}}\)[/tex], we need to apply the properties of exponents and radicals. Here’s a step-by-step solution:
1. Express the expression under a single exponent:
[tex]\[
\sqrt[3]{128 x^{12} y^{13} z^{16}} = (128 x^{12} y^{13} z^{16})^{1/3}
\][/tex]
2. Separate the expression into its individual components, each raised to the power of [tex]\(1/3\)[/tex]:
[tex]\[
(128 x^{12} y^{13} z^{16})^{1/3} = 128^{1/3} \cdot (x^{12})^{1/3} \cdot (y^{13})^{1/3} \cdot (z^{16})^{1/3}
\][/tex]
3. Calculate the cube root of each component separately:
- The cube root of [tex]\(128\)[/tex]:
[tex]\[
128^{1/3} \approx 5.039684199579492
\][/tex]
- Simplify the exponent of [tex]\(x\)[/tex]:
[tex]\[
(x^{12})^{1/3} = x^{12 \cdot \frac{1}{3}} = x^4
\][/tex]
- Simplify the exponent of [tex]\(y\)[/tex]:
[tex]\[
(y^{13})^{1/3} = y^{13 \cdot \frac{1}{3}} \approx y^{4.333333333333333}
\][/tex]
- Simplify the exponent of [tex]\(z\)[/tex]:
[tex]\[
(z^{16})^{1/3} = z^{16 \cdot \frac{1}{3}} \approx z^{5.333333333333333}
\][/tex]
4. Combine all the simplified components:
[tex]\[
\sqrt[3]{128 x^{12} y^{13} z^{16}} = 5.039684199579492 \cdot x^4 \cdot y^{4.333333333333333} \cdot z^{5.333333333333333}
\][/tex]
Therefore, the fully simplified form of [tex]\(\sqrt[3]{128 x^{12} y^{13} z^{16}}\)[/tex] is:
[tex]\[
5.039684199579492 \cdot x^4 \cdot y^{4.333333333333333} \cdot z^{5.333333333333333}
\][/tex]
