Certainly! Let's find the value of [tex]\(\sqrt[3]{(7d)^{22}}\)[/tex] step-by-step.
1. Start with the given expression:
[tex]\[
\sqrt[3]{(7d)^{22}}
\][/tex]
2. Apply the property of exponents to rewrite the expression inside the cube root:
[tex]\[
(7d)^{22} = 7^{22} \cdot d^{22}
\][/tex]
3. Now, take the cube root of the entire expression:
[tex]\[
\sqrt[3]{7^{22} \cdot d^{22}}
\][/tex]
4. Use the property of roots to distribute the cube root over the multiplication:
[tex]\[
\sqrt[3]{7^{22}} \cdot \sqrt[3]{d^{22}}
\][/tex]
5. Simplify each part separately:
- For [tex]\(\sqrt[3]{7^{22}}\)[/tex], you use the rule [tex]\(\sqrt[3]{a^b} = a^{b/3}\)[/tex]:
[tex]\[
\sqrt[3]{7^{22}} = 7^{22/3}
\][/tex]
- For [tex]\(\sqrt[3]{d^{22}}\)[/tex], the same rule applies:
[tex]\[
\sqrt[3]{d^{22}} = d^{22/3}
\][/tex]
6. Combine the simplified parts:
[tex]\[
\sqrt[3]{(7d)^{22}} = 7^{22/3} \cdot d^{22/3}
\][/tex]
7. Calculate the numerical value of [tex]\(7^{22/3}\)[/tex]:
By approximation, we obtain:
[tex]\[
7^{22/3} \approx 1575381.08505392
\][/tex]
8. Combine the results:
[tex]\[
1575381.08505392 \cdot d^{22/3}
\][/tex]
So, the final simplified expression is:
[tex]\[
\sqrt[3]{(7d)^{22}} = 1575381.08505392 \cdot d^{22/3}
\][/tex]
