O P Malhotra, S K Gupta and, Anubhuti Gangal Solutions for Exercise 3: Exercise

If $2205=3^a\times 5^b\times 7^c$, find the numerical values of $a, b$ and $c$.

Prove that: $\frac{a+b+c}{a^{-1}{b}^{-1}+b^{-1}{c}^{-1}+c^{-1}{a}^{-1}}=abc$

Prove that:  $\frac{1}{1+x^{b-a}+x^{c-a}}+\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{b-c}+x^{a-c}}=1$

Prove that: $\sqrt[ab]{\frac{x^a}{x^b}}.\sqrt[bc]{\frac{x^b}{x^c}}.\sqrt[ca]{\frac{x^c}{x^a}}=1$

Simplify: $\left(p^{\frac{1}{3}}-p^{\frac{-1}{3}}\right)\left(p^{\frac{2}{3}}+1+p^{\frac{-2}{3}}\right)$

Simplify: ${\left(\sqrt{11}+\sqrt{3}\right)}^{\frac{1}{3}}{\left(\sqrt{11}-\sqrt{3}\right)}^{\frac{1}{3}}$

If $3^x=5^y={45}^z$, prove that $x=\frac{2yz}{y-z}$.

If $x=a{b}^{p-1}, y=a{b}^{q-1}$ and $z=a{b}^{r-1}$, prove that $x^{q-r}\times y^{r-p}.z^{p-q}=1$.