To solve the expression [tex]\(\sqrt[3]{d^{11}}\)[/tex], we'll break it down into a few clear steps:
1. Understand the expression: The expression [tex]\(\sqrt[3]{d^{11}}\)[/tex] is asking for the cube root of [tex]\(d^{11}\)[/tex], which can also be written as taking [tex]\(d^{11}\)[/tex] to the power of [tex]\(\frac{1}{3}\)[/tex].
2. Rewrite the radical as an exponent: A cube root can be expressed as a fractional exponent. Thus, [tex]\(\sqrt[3]{d^{11}}\)[/tex] can be rewritten as [tex]\((d^{11})^{\frac{1}{3}}\)[/tex].
3. Apply the laws of exponents: Using the power of a power rule in exponents, which states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify [tex]\((d^{11})^{\frac{1}{3}}\)[/tex] to [tex]\(d^{11 \cdot \frac{1}{3}}\)[/tex].
4. Multiply the exponents: You need to multiply [tex]\(11\)[/tex] by [tex]\(\frac{1}{3}\)[/tex], resulting in [tex]\(\frac{11}{3}\)[/tex].
5. Write the final result: After performing the multiplication, the expression simplifies to [tex]\(d^{\frac{11}{3}}\)[/tex].
So, the simplified form of [tex]\(\sqrt[3]{d^{11}}\)[/tex] is [tex]\((d^{11})^{\frac{1}{3}} = d^{\frac{11}{3}}\)[/tex].
