PRACTICE QUESTIONS CLASSROOM WING

Which of the following is greater between the two numbers?

$2^{21}\text{ and }{3}^{14}$

If $5^x=3^y={45}^z$, then prove that $\frac{1}{z}=\frac{1}{x}+\frac{2}{y}.$

Prove that $\frac{1}{1+p+q^{-1}}+\frac{1}{1+q+r^{-1}}+\frac{1}{1+r+p^{-1}}=1$ of $pqr=1$.

Simplify the given expression.

$\frac{(a+b+c{)}^{a+b}(a+b+c{)}^{b+c}(a+b+c{)}^{c+a}}{{\left[(a+b+c{)}^a(a+b+c{)}^b(a+b+c{)}^c\right]}^2}$

Simplify the given expression.

${\left(a^{p+q}\right)}^{p-q}{\left(a^{q+r}\right)}^{q-r}{\left(a^{r+p}\right)}^{r-p}$

Simplify the given expression.

${\left[{\left(\frac{a}{b}\right)}^{\sqrt{p}+\sqrt{q}}\right]}^{\sqrt{p}-\sqrt{q}}{\left[{\left(\frac{a}{b}\right)}^{\sqrt{q}+\sqrt{r}}\right]}^{\sqrt{q}-\sqrt{r}}{\left[{\left(\frac{a}{b}\right)}^{\sqrt{r}+\sqrt{p}}\right]}^{\sqrt{r}-\sqrt{p}}$

Prove that ${\left(\frac{x^{-1}+y^{-1}}{x^{-1}}\right)}^{-1}+{\left(\frac{x^{-1}-y^{-1}}{x^{-1}}\right)}^{-1}=\frac{2y^2}{y^2-x^2}$

If $a, b$ and $c$ are real numbers, than show that $\sqrt[5]{a{b}^{-1}}\times \sqrt[5]{b{c}^{-1}}\times \sqrt[5]{c{a}^{-1}}=1$.