Question

which answer is equivalent to \\(\frac{sqrt{25}}{sqrt{30}}\\)?

a. \\(sqrt{\frac{25}{30}}\\)

b. \\(\frac{25}{30}\\)

c. \\(\frac{sqrt{5}}{sqrt{6}}\\)

d. \\(\frac{sqrt3{25}}{sqrt3{30}}\\)

Explanation:

Step1: Recall the quotient rule for square roots

The quotient rule for square roots states that $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ for $a\geq0$ and $b > 0$.

Step2: Apply the rule to the given expression

Given the expression $\frac{\sqrt{25}}{\sqrt{30}}$, by the quotient rule for square roots, we have $\frac{\sqrt{25}}{\sqrt{30}}=\sqrt{\frac{25}{30}}$. We can also simplify $\sqrt{\frac{25}{30}}$ by simplifying the fraction $\frac{25}{30}=\frac{5}{6}$, so $\sqrt{\frac{25}{30}}=\sqrt{\frac{5}{6}}=\frac{\sqrt{5}}{\sqrt{6}}$ (by the quotient rule again: $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$). But let's check the options. Option A is $\sqrt{\frac{25}{30}}$, which is equivalent to $\frac{\sqrt{25}}{\sqrt{30}}$ by the quotient rule. Let's verify each option:

• Option A: $\sqrt{\frac{25}{30}}$ is equal to $\frac{\sqrt{25}}{\sqrt{30}}$ by the quotient rule for square roots ($\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$).
• Option B: $\frac{25}{30}$ is a rational number, while $\frac{\sqrt{25}}{\sqrt{30}}=\frac{5}{\sqrt{30}}$ which is irrational, so they are not equivalent.
• Option C: $\frac{\sqrt{5}}{\sqrt{6}}$ is equal to $\sqrt{\frac{5}{6}}$, and $\frac{\sqrt{25}}{\sqrt{30}}=\sqrt{\frac{25}{30}}=\sqrt{\frac{5}{6}}=\frac{\sqrt{5}}{\sqrt{6}}$, but let's check the first equivalent form. Wait, the question is which answer is equivalent. The first equivalent form from the quotient rule is $\sqrt{\frac{25}{30}}$ (Option A). Also, simplifying $\frac{25}{30}$ to $\frac{5}{6}$, so $\sqrt{\frac{25}{30}}=\sqrt{\frac{5}{6}}=\frac{\sqrt{5}}{\sqrt{6}}$, but let's check the options. Wait, the original expression is $\frac{\sqrt{25}}{\sqrt{30}}$. Let's compute the numerical value: $\frac{\sqrt{25}}{\sqrt{30}}=\frac{5}{\sqrt{30}}\approx\frac{5}{5.477}\approx0.913$. $\sqrt{\frac{25}{30}}=\sqrt{\frac{5}{6}}\approx\sqrt{0.833}\approx0.913$, so they are equal. $\frac{25}{30}\approx0.833

eq0.913$, so B is wrong. $\frac{\sqrt{5}}{\sqrt{6}}\approx\frac{2.236}{2.449}\approx0.913$, which is equal, but let's check the options. Wait, the options: A is $\sqrt{\frac{25}{30}}$, C is $\frac{\sqrt{5}}{\sqrt{6}}$. But the first equivalent form from the quotient rule is A. Wait, actually, the quotient rule says $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$, so $\frac{\sqrt{25}}{\sqrt{30}}=\sqrt{\frac{25}{30}}$, which is Option A. Also, $\sqrt{\frac{25}{30}}=\sqrt{\frac{5}{6}}=\frac{\sqrt{5}}{\sqrt{6}}$, which is Option C. But let's check the problem again. Wait, the original expression is $\frac{\sqrt{25}}{\sqrt{30}}$. Let's see the options. Option A is $\sqrt{\frac{25}{30}}$, which is directly equivalent by the quotient rule. Option C is also equivalent, but let's check the options. Wait, maybe I made a mistake. Wait, $\frac{\sqrt{25}}{\sqrt{30}}=\frac{5}{\sqrt{30}}$, and $\sqrt{\frac{25}{30}}=\sqrt{\frac{5}{6}}=\frac{\sqrt{5}}{\sqrt{6}}=\frac{5}{\sqrt{30}}$ (rationalizing the denominator: $\frac{\sqrt{5}}{\sqrt{6}}=\frac{\sqrt{5}\times\sqrt{6}}{\sqrt{6}\times\sqrt{6}}=\frac{\sqrt{30}}{6}$, and $\frac{5}{\sqrt{30}}=\frac{5\sqrt{30}}{30}=\frac{\sqrt{30}}{6}$, so they are equal). But the question is which answer is equivalent. Let's check the options again. The first equivalent form from the quotient rule is Option A: $\sqrt{\frac{25}{30}}$. Let's confirm with the rule: $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$, so $\frac{\sqrt{25}}{\sqrt{30}}=\sqrt{\frac{25}{30}}$, so Option A is equivalent.

Answer:

A. $\sqrt{\frac{25}{30}}$