Question

Simplify the following expressions.
Classify the following numbers as rational or irrational.

• A. $(5+\sqrt{7})(2+\sqrt{5})$
• B. $(5+\sqrt{5})(5-\sqrt{5})$
• C. ${(\sqrt{3}+\sqrt{7})}^2$
• D. $(\sqrt{11}-\sqrt{7})(\sqrt{11}+\sqrt{7})$

Answer 1

The correct option is D $(\sqrt{11}-\sqrt{7})(\sqrt{11}+\sqrt{7})$

$A:$

$\left(5+\sqrt{7}\right)\left(2+\sqrt{5}\right)$

$=10+5\sqrt{5}+2\sqrt{7}+\sqrt{35}$

Now, $10$ is rational and $\sqrt{5},\sqrt{7}$ are non terminating , non repeating is an irrational

and we know that $rational+irrational=irrational$

Therefore, $\left(5+\sqrt{7}\right)\left(2+\sqrt{5}\right)$ is irrational

$B:$

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

$=5^2+{\left(\sqrt{5}\right)}^2=25-5$

$=5$, which is rational

So, $\left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right)$

Is rational number.

$C:$

${\left(\sqrt{3}+\sqrt{7}\right)}^2$

$={\left(\sqrt{3}\right)}^2+{\left(\sqrt{7}\right)}^2+2\sqrt{3}\sqrt{7}$

$={\left(\sqrt{3}\right)}^2+{\left(\sqrt{7}\right)}^2+2\sqrt{21}=3+7+2\sqrt{21}=10+2\sqrt{21}$

and $10$ and $\sqrt{21}$ are both rational.

Therefore, ${\left(\sqrt{3}+\sqrt{7}\right)}^2$ is rational.

$D:$

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

$={\left(11\right)}^2-{\left(\sqrt{7}\right)}^2$

$=11-7=4$, which is rational.

Therefore $\left(11-\sqrt{7}\right)\left(11+\sqrt{7}\right)$ is rational.