Question

simplify the expression: ( 20 \times \frac{3sqrt{2}}{7sqrt{12}} )

Explanation:

Step1: Simplify the square root in the denominator

We know that $\sqrt{12}=\sqrt{4\times3}=\sqrt{4}\times\sqrt{3} = 2\sqrt{3}$. So we can rewrite the fraction $\frac{3\sqrt{2}}{7\sqrt{12}}$ as $\frac{3\sqrt{2}}{7\times2\sqrt{3}}=\frac{3\sqrt{2}}{14\sqrt{3}}$.

Step2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by $\sqrt{3}$. So we have $\frac{3\sqrt{2}\times\sqrt{3}}{14\sqrt{3}\times\sqrt{3}}$. Since $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$, the numerator becomes $3\sqrt{6}$ and the denominator becomes $14\times3 = 42$. So the fraction simplifies to $\frac{3\sqrt{6}}{42}=\frac{\sqrt{6}}{14}$.

Step3: Multiply by 20

Now we multiply this result by 20: $20\times\frac{\sqrt{6}}{14}=\frac{20\sqrt{6}}{14}=\frac{10\sqrt{6}}{7}$.

Answer:

$\frac{10\sqrt{6}}{7}$