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

Solve the following exercise:

$\frac{\sqrt{x^4}}{x}=$

Express the definition of root as a power:

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

Remember that in a square root (also called "root to the power of 2") we don't write the root's power as shown below:

$n=2$

Meaning:

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

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

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

Let's recall the power law for a power of a power:

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

Apply this law to the numerator of the fraction in the expression that we obtained in the last step:

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

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,

Continue to simplify the expression that we obtained. Begin by reducing the fraction with the power exponent in the numerator term and then proceed to apply the power law for division between terms with identical bases:

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

Simplify the fraction in the now complete expression:

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

Let's summarize the various steps of the solution that we obtained: As shown below

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

Therefore the correct answer is answer A.