Question

1. select the equivalent expressions.

\\(\sqrt{\dfrac{150}{25}}\\)

a. \\(\sqrt{2} \cdot \sqrt{3}\\)
b. \\(\sqrt{\dfrac{1}{6}}\\)
c. \\(\sqrt{6}\\)
d. 3
e. \\(\dfrac{1}{\sqrt{6}}\\)

Explanation:

Step1: Simplify the given expression

First, simplify $\sqrt{\frac{150}{25}}$. We know that $\frac{150}{25} = 6$, so $\sqrt{\frac{150}{25}}=\sqrt{6}$.

Step2: Analyze option a

Simplify $\sqrt{2}\cdot\sqrt{3}$. Using the property of square roots $\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$ (where $a\geq0$ and $b\geq0$), we have $\sqrt{2}\cdot\sqrt{3}=\sqrt{2\times3}=\sqrt{6}$.

Step3: Analyze option b

Simplify $\sqrt{\frac{1}{6}}$. We can rewrite this as $\frac{\sqrt{1}}{\sqrt{6}}=\frac{1}{\sqrt{6}}$, which is not equal to $\sqrt{6}$.

Step4: Analyze option c

Option c is $\sqrt{6}$, which is equal to our simplified original expression.

Step5: Analyze option d

Option d is $3$, and $\sqrt{6}\approx2.45
eq3$, so it is not equivalent.

Step6: Analyze option e

Option e is $\frac{1}{\sqrt{6}}$, which is the reciprocal of $\sqrt{6}$ (after rationalizing, but even without rationalizing, it's not equal to $\sqrt{6}$), so it is not equivalent.

Answer:

a. $\boldsymbol{\sqrt{2}\cdot\sqrt{3}}$, c. $\boldsymbol{\sqrt{6}}$