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.:

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

We'll continue, since there is 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 and perform (separately) the addition of fractions in the exponent of the expression we got, this will be done by expanding each of the fractions to the common denominator - the number 6 (which is the smallest common denominator), 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 reduced the resulting fraction,

Let's return to the problem and substitute the result of the fraction addition, meaning we get:

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

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.