Question

select the expression that is not equivalent to \\(\frac{\sqrt{136}}{\sqrt{2}}\\).
\\(\circ\\) \\(\sqrt{\frac{136}{2}}\\)
\\(\circ\\) \\(\sqrt{68}\\)
\\(\circ\\) \\(\frac{\sqrt{100}+\sqrt{36}}{\sqrt{2}}\\)
\\(\circ\\) \\(2\sqrt{17}\\)

Explanation:

Step1: Recall the quotient rule of square roots

The quotient rule states that $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ for $a\geq0$ and $b>0$. So, $\frac{\sqrt{136}}{\sqrt{2}}=\sqrt{\frac{136}{2}}$, so the first option is equivalent.

Step2: Simplify $\sqrt{\frac{136}{2}}$

Calculate $\frac{136}{2} = 68$, so $\sqrt{\frac{136}{2}}=\sqrt{68}$, so the second option is equivalent.

Step3: Analyze the third option

Simplify the numerator of the third option: $\sqrt{100}+\sqrt{36}=10 + 6=16$. So the third option becomes $\frac{16}{\sqrt{2}}$. Now, simplify $\frac{\sqrt{136}}{\sqrt{2}}=\frac{\sqrt{4\times34}}{\sqrt{2}}=\frac{2\sqrt{34}}{\sqrt{2}}=\sqrt{2}\times\sqrt{34}=\sqrt{68}\approx8.246$, and $\frac{16}{\sqrt{2}}=\frac{16\sqrt{2}}{2} = 8\sqrt{2}\approx11.313$, which are not equal.

Step4: Simplify $\sqrt{68}$

Factor $68 = 4\times17$, so $\sqrt{68}=\sqrt{4\times17}=2\sqrt{17}$, so the fourth option is equivalent.

Answer:

$\frac{\sqrt{100}+\sqrt{36}}{\sqrt{2}}$ (the third option)