Solve the Expression: Multiplying ⁶√7 × ²√7 Step-by-Step

Solve the following exercise:

$\sqrt[6]{7}\cdot\sqrt[2]{7}=\text{ }$

In order to simplify the given expression, we will apply two laws of exponents:

a. Root definition as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

b. Law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Begin by converting the roots to exponents using the law mentioned in a:

$\sqrt[\textcolor{red}{6}]{7}\cdot\sqrt{7}= \\ \downarrow\\ 7^{\frac{1}{\textcolor{red}{6}}}\cdot7^{\frac{1}{\textcolor{blue}{2}}}=$

Given that there is a multiplication operation present between the two terms with identical bases, we'll apply the law of exponents mentioned in b:

$7^{\frac{1}{6}}\cdot7^{\frac{1}{2}}= \\ 7^{\frac{1}{6}+\frac{1}{2}}=$

We'll continue by performing the addition of fractions in the exponent of the expression separately. This is achieved by expanding each of the fractions to the common denominator—the number 6, which is the smallest common denominator—and then we'll perform the multiplication and addition operations in the fraction numerator:

$\frac{1}{6}+\frac{1}{2}=\\ \frac{1\cdot1+1\cdot3}{6}=\\ \frac{1+3}{6}=\\ \frac{\not{4}}{\not{6}}=\\ \frac{2}{3}$

In the final step, we reduce the resulting fraction.

Let's return to the problem and substitute in the result of the fraction addition:

$7^{\frac{1}{6}+\frac{1}{2}}=\\ \boxed{7^{\frac{2}{3}}}$

Let's summarize the expression simplification process:

$\sqrt[6]{7}\cdot\sqrt{7}= \\ \downarrow\\ 7^{\frac{1}{6}}\cdot7^{\frac{1}{2}}= \\ 7^{\frac{1}{6}+\frac{1}{2}}=\\ \boxed{7^{\frac{2}{3}}}$

Therefore, the correct answer is answer c.