Operations on Real Numbers

Simplify : $\left(\sqrt{11}-\sqrt{7}\right)\left(\sqrt{11}+\sqrt{7}\right)$

Simplify : $\left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right)$

If the quotient obtained on dividing $16\sqrt{15\mathrm{}}\mathrm{}$  by   $4\sqrt{3}$ is $4\sqrt{k}$, then find the value of $k$. 

Find the product of $2\sqrt{5}\mathrm{}$ and $6\sqrt{5}$.

If the value of $\frac{1}{\sqrt{7}-1}=\frac{\sqrt{7}+1}{k}$ then find the value of k.

Multiply $6\sqrt{3}$ by $13\sqrt{3}$

The product of two irrational numbers is irrational. (Write answer Yes or No).

Simplify: $\left(7\sqrt{2}\right)\times \left(2\sqrt{7}\right)=p\sqrt{q}$. Find the value of $q$.

$3÷\frac{17}{13}=\frac{3}{1}÷\frac{17}{13}=$ _____ $=$ _____ $=$ _____

Is it a rational number?

$-\frac{2}{3}÷\frac{6}{11}=$ _____ $=$ _____ $=$ _____

Is it a rational number?

$\frac{2}{1}\times \frac{19}{13}=$ _____

Check whether their product is a rational number or not?

$\frac{3}{7}\times \frac{5}{2}=$ _____

Check whether their product is a rational number or not?

$\left(\frac{48}{9}\right)-\frac{11}{8}=$ _____

Is this a rational number?

$\frac{2}{3}-\frac{3}{7}=\frac{14-9}{21}=$ _____ 

Is it a rational number?

$\frac{2}{7}+\frac{-2}{7}=$ _____

Will the result be a rational number?

$8+\left(\frac{-19}{2}\right)=$ _____ 

Is it a rational number?

If $x = \frac{1}{7+4\sqrt{3}} and y=\frac{1}{7-4\sqrt{3}}, find xy$

If $x=\frac{1}{7+4\sqrt{3}} and y=\frac{1}{7-4\sqrt{3}}, find x+y$

If $a = \frac{\sqrt{2}+1}{\sqrt{2}-1}\text{and} b = \frac{\sqrt{2}-1}{\sqrt{2}+1}$, then find the value of $a^2+b^2+ab$.