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

In order to simplify the given expression, we will use 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:

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

Let's start by converting the roots to exponents using the law mentioned in a:

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

Since there is a multiplication between two terms with identical bases, we'll use 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 separately performing the addition of fractions in the exponent of the expression. This is done 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.