Simplify the Square Root Expression: √(5x⁴)

To simplify the given expression, we will use 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:

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

Let's start with converting the fourth root to an exponent using the law of exponents mentioned in a:

$\sqrt{5x^4}= \\ \downarrow\\ (5x^4)^{\frac{1}{2}}=$

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

$(5x^4)^{\frac{1}{2}}= \\ 5^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}}$

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 multiplication):

$5^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}} = \\ 5^{\frac{1}{2}}\cdot x^{4\cdot\frac{1}{2}}=\\ 5^{\frac{1}{2}}\cdot x^{2}=\\ \boxed{\sqrt{5}x^2}$

In the final step, we converted the one-half power applied to the first term in the multiplication back to the 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.