To find the product [tex]\(\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d}\)[/tex], we start by recalling what the cube root operation represents. The cube root of a number [tex]\(d\)[/tex], denoted [tex]\(\sqrt[3]{d}\)[/tex], is a number which, when raised to the power of 3, equals [tex]\(d\)[/tex].
Mathematically, the cube root of [tex]\(d\)[/tex] can be written as [tex]\(d^{\frac{1}{3}}\)[/tex]. Now, let's express the given product in terms of exponents:
[tex]\[
\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d} = d^{\frac{1}{3}} \cdot d^{\frac{1}{3}} \cdot d^{\frac{1}{3}}
\][/tex]
When multiplying terms with the same base, we add the exponents:
[tex]\[
d^{\frac{1}{3}} \cdot d^{\frac{1}{3}} \cdot d^{\frac{1}{3}} = d^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}}
\][/tex]
Next, we sum the exponents:
[tex]\[
\frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{3}{3} = 1
\][/tex]
Therefore,
[tex]\[
d^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = d^1 = d
\][/tex]
So, the product [tex]\(\sqrt[3]{d} \cdot \sqrt[3]{d} \cdot \sqrt[3]{d}\)[/tex] equals [tex]\(d\)[/tex]. The correct answer is:
[tex]\[ \boxed{d} \][/tex]
