Solve √(16x²): Simplifying Square Root Expression Step-by-Step

Solve the following exercise:

$\sqrt{16x^2}=$

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:

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

We'll start by converting the fourth root to an exponent using the law of exponents mentioned in a.:

$\sqrt{16x^2}= \\ \downarrow\\ (16x^2)^{\frac{1}{2}}=$

We'll continue, using the law of exponents mentioned in b. and apply the exponent to each factor in the parentheses:

$(16x^2)^{\frac{1}{2}}= \\ 16^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}}$

We'll once again continue, using the law of exponents mentioned in c. and perform the exponent applied to the term with an exponent in parentheses (the second factor in the multiplication):

$16^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}} = \\ 16^{\frac{1}{2}}\cdot x^{2\cdot\frac{1}{2}}=\\ 16^{\frac{1}{2}}\cdot x^{1}=\\ \sqrt{16}\cdot x=\\ \boxed{4x}$

In the final steps, first we converted the power of one-half applied to the first factor in the multiplication back to the fourth root form, again, according to the definition of root as an exponent mentioned in a. (in the opposite direction) and then we calculated the known fourth root of 16.

Therefore, the correct answer is answer d.