Question

1. \\(\frac{5sqrt{6}}{2sqrt{75}} =\\)

Explanation:

Step1: Simplify $\sqrt{75}$

We know that $75 = 25\times3$, so $\sqrt{75}=\sqrt{25\times3}=\sqrt{25}\times\sqrt{3}=5\sqrt{3}$ (since $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ for $a\geq0,b\geq0$ and $\sqrt{25} = 5$).

Step2: Substitute $\sqrt{75}$ into the fraction

The original fraction is $\frac{5\sqrt{6}}{2\sqrt{75}}$, substituting $\sqrt{75}=5\sqrt{3}$ we get $\frac{5\sqrt{6}}{2\times5\sqrt{3}}$.

Step3: Simplify the fraction

The $5$ in the numerator and the $5$ in the denominator (from $2\times5\sqrt{3}$) cancel out, so we have $\frac{\sqrt{6}}{2\sqrt{3}}$.

Step4: Rationalize the denominator

Multiply the numerator and the denominator by $\sqrt{3}$: $\frac{\sqrt{6}\times\sqrt{3}}{2\sqrt{3}\times\sqrt{3}}$. Using the property $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$, the numerator becomes $\sqrt{18}$ and the denominator becomes $2\times3 = 6$.

Step5: Simplify $\sqrt{18}$

Since $18=9\times2$, $\sqrt{18}=\sqrt{9\times2}=\sqrt{9}\times\sqrt{2}=3\sqrt{2}$. So now the fraction is $\frac{3\sqrt{2}}{6}$.

Step6: Simplify the final fraction

We can divide both the numerator and the denominator by $3$, so $\frac{3\sqrt{2}\div3}{6\div3}=\frac{\sqrt{2}}{2}$.

Answer:

$\frac{\sqrt{2}}{2}$