what is the following product?\n$sqrt4{7}cdotsqrt4{7}cdotsqrt4{7}cdotsqrt4{7}$\n$4(sqrt4{7})$\n$4sqrt{7}$\n$7…

Explanation:

Step1: Use exponent - multiplication rule

When multiplying numbers with the same base and the same root, we add the exponents. For $\sqrt[4]{7}\cdot\sqrt[4]{7}\cdot\sqrt[4]{7}\cdot\sqrt[4]{7}$, the base is 7 and the index of the radical is 4. Each $\sqrt[4]{7}$ can be written as $7^{\frac{1}{4}}$. So we have $7^{\frac{1}{4}}\cdot7^{\frac{1}{4}}\cdot7^{\frac{1}{4}}\cdot7^{\frac{1}{4}}$. According to the rule $a^m\cdot a^n=a^{m + n}$, we get $7^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}$.

Step2: Calculate the sum of exponents

$\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{1 + 1+1 + 1}{4}=\frac{4}{4}=1$. So $7^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}=7^1 = 7$.

Answer:

7