Simplify the Expression: √(x⁴)/x Step-by-Step Solution

Let's use the definition of root as a power:

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

when we remember that in a square root (also called "root to the power of 2") we don't write the root's power and:

$n=2$

meaning:

$\sqrt{a}=\sqrt[2]{a}=a^{\frac{1}{2}}$

Let's return to the problem and convert using the root definition we mentioned above the root in the numerator of the fraction in the problem:

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

Now let's remember the power law for power of power:

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

Let's apply this law to the numerator of the fraction in the expression we got in the last step:

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

where in the first step we applied the above power law and in the second step we performed the multiplication in the power exponent of the numerator term,

Let's continue and simplify the expression we got, first we'll reduce the fraction with the power exponent in the numerator term and then we'll use the power law for division between terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

to simplify the fraction in the complete expression:

$\frac{x^\frac{4}{2}}{x}=\frac{x^2}{x}=x^{2-1}=x$

Let's summarize the solution steps, we got that:

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

Therefore the correct answer is answer A.