Question

express in simplest form with a rational denominator.
\\(\dfrac{7}{\sqrt{40}}\\)

Explanation:

Step1: Simplify the square root in the denominator

First, we simplify $\sqrt{40}$. We can factor 40 as $4\times10$, and since $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ (for $a\geq0, b\geq0$), we have $\sqrt{40}=\sqrt{4\times10}=\sqrt{4}\times\sqrt{10}=2\sqrt{10}$. So the fraction becomes $\frac{7}{2\sqrt{10}}$.

Step2: Rationalize the denominator

To rationalize the denominator (i.e., eliminate the square root from the denominator), we multiply the numerator and the denominator by $\sqrt{10}$. This is because when we multiply $\sqrt{10}$ by itself, we get 10, which is a rational number. So we have:
\[
\frac{7\times\sqrt{10}}{2\sqrt{10}\times\sqrt{10}}
\]

Step3: Simplify the denominator

Simplify the denominator: $2\sqrt{10}\times\sqrt{10}=2\times(\sqrt{10})^2 = 2\times10 = 20$. The numerator is $7\sqrt{10}$. So the fraction becomes $\frac{7\sqrt{10}}{20}$.

Answer:

$\frac{7\sqrt{10}}{20}$