what is the following product? assume $dgeq0$.\n$sqrt3{d}cdotsqrt3{d}cdotsqrt3{d}$\n$d$\n$d^{3}$\n$3(sqrt3{d}…

Explanation:

Step1: Use exponent - rule for radicals

The cube - root $\sqrt[3]{d}$ can be written as $d^{\frac{1}{3}}$. So, $\sqrt[3]{d}\cdot\sqrt[3]{d}\cdot\sqrt[3]{d}=d^{\frac{1}{3}}\cdot d^{\frac{1}{3}}\cdot d^{\frac{1}{3}}$.

Step2: Apply the product rule of exponents

When multiplying terms with the same base ($a^m\cdot a^n=a^{m + n}$), for $a = d$, $m=\frac{1}{3}$, and $n=\frac{1}{3}$, we have $d^{\frac{1}{3}}\cdot d^{\frac{1}{3}}\cdot d^{\frac{1}{3}}=d^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}$.

Step3: Calculate the sum of the exponents

$\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=\frac{1 + 1+1}{3}=\frac{3}{3}=1$. So, $d^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}=d^1=d$.

Answer:

A. $d$