Solve the Square Root Expression: Simplifying √(11x²)

Solve the following exercise:

$\sqrt{11x^2}=$

In order to simplify the given expression, apply the following three laws of exponents:

a. Definition of root as an exponent:

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

b. Law of exponents for an exponent applied to terms in parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

c. Law of exponents for an exponent raised to an exponent:

$(a^m)^n=a^{m\cdot n}$

Begin by converting the fourth root to an exponent using the law of exponents mentioned in a:

$\sqrt{11x^2}= \\ \downarrow\\ (11x^2)^{\frac{1}{2}}=$

Next, we'll use the law of exponents mentioned in b and apply the exponent to each term in the parentheses:

$(11x^2)^{\frac{1}{2}}= \\ 11^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}}$

Let's continue, using the law of exponents mentioned in c and perform the exponent applied to the term with an exponent in parentheses (the second term in the multiplication):

$11^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}} = \\ 11^{\frac{1}{2}}\cdot x^{2\cdot\frac{1}{2}}=\\ 11^{\frac{1}{2}}\cdot x^{1}=\\ \boxed{\sqrt{11} x}\\$In the final step, we converted the one-half exponent applied to the first term in the multiplication back to a fourth root, again, according to the definition of root as an exponent mentioned in a (in the reverse direction).

Therefore, the correct answer is answer c.