Question

use the quotient rule to simplify. assume all variables represent non-negative values.\sqrt{\frac{7x^5}{112x}}\sqrt{\frac{7x^5}{112x}} = \square\\(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for

Explanation:

Step1: Apply Quotient Rule for Radicals

The quotient rule for radicals states that $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ (for $a\geq0, b > 0$). Also, we can simplify the fraction inside the radical first. Simplify $\frac{7x^{5}}{112x}$.

First, simplify the coefficients: $\frac{7}{112}=\frac{1}{16}$ (dividing numerator and denominator by 7). Then, simplify the variables using the quotient rule for exponents: $\frac{x^{5}}{x}=x^{5 - 1}=x^{4}$ (since $a^{m}/a^{n}=a^{m - n}$ for $a
eq0$). So the fraction inside the radical becomes $\frac{x^{4}}{16}$.

Now our expression is $\sqrt{\frac{x^{4}}{16}}$.

Step2: Simplify the Radical

We know that $\sqrt{\frac{x^{4}}{16}}=\frac{\sqrt{x^{4}}}{\sqrt{16}}$ (by quotient rule for radicals).

Simplify $\sqrt{x^{4}}$: since $x$ is non - negative, $\sqrt{x^{4}}=x^{2}$ (because $(x^{2})^{2}=x^{4}$). And $\sqrt{16} = 4$.

So $\frac{\sqrt{x^{4}}}{\sqrt{16}}=\frac{x^{2}}{4}$.

Answer:

$\frac{x^{2}}{4}$