Simplify the Square Root Expression: √(12x⁴)

Solve the following exercise:

$\sqrt{12x^4}=$

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

$\sqrt{12x^4}= \\ \downarrow\\ (12x^4)^{\frac{1}{2}}=$

Next, apply the law of exponents mentioned in b. and proceed to apply the exponent to each factor inside of the parentheses:

$(12x^4)^{\frac{1}{2}}= \\ 12^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}}$

Let's continue, using the law of exponents mentioned in c. and perform the exponent operation on the term with an exponent in the parentheses (the second term in the product):

$12^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}} = \\ 12^{\frac{1}{2}}\cdot x^{4\cdot\frac{1}{2}}=\\ 12^{\frac{1}{2}}\cdot x^{2}=\\ \boxed{\sqrt{12} x^2}=\\$In the final step, we converted the one-half exponent on the first term in the product back to a fourth root form, again, according to the definition of root as an exponent mentioned in a. (in the reverse direction).

Therefore, the correct answer is answer c.