R. D. Sharma Solutions for Chapter: Exponents of Real Numbers, Exercise 2: EXERCISE

Simplify:

${\left(\frac{\sqrt{2}}{5}\right)}^8÷{\left(\frac{\sqrt{2}}{5}\right)}^{13}$

If $3^{4x}=(81{)}^{-1}$ and ${10}^{\frac{1}{y}}=0.0001$, find the value of $2^{-x+4y}$.

Solve the following equation:

$3^{x+1}=27\times 3^4$

Solve the following equation:

$4^{x-1}\times (0.5{)}^{3-2x}={\left(\frac{1}{8}\right)}^x$

Solve the following equation:

$\sqrt{\frac{a}{b}}={\left(\frac{b}{a}\right)}^{1-2x},$  where $a, b$ are distinct positive primes.

If $a$ and $b$ are distinct positive primes such that

$(a+b{)}^{-1}\left(a^{-1}+b^{-1}\right)=a^x{b}^y,$ find $x+y+2.$

Simplify:

${\left(\frac{x^{a+b}}{x^c}\right)}^{a-b}{\left(\frac{x^{b+c}}{x^a}\right)}^{b-c}{\left(\frac{x^{c+a}}{x^b}\right)}^{c-a}$

Simplify:

$\sqrt[lm]{\frac{x^l}{x^m}}\times \sqrt[mn]{\frac{x^m}{x^n}}\times \sqrt[nl]{\frac{x^n}{x^l}}$.