RD Sharma solutions for Mathematics [English] Class 9 chapter 2 - Exponents of Real Numbers [Latest edition]

Simplify the following

${\displaystyle 3{\left(a^4{b}^3\right)}^{10}\times 5{\left(a^2{b}^2\right)}^3}$

Simplify the following

${\displaystyle {\left(2x^{-2}{y}^3\right)}^3}$

Simplify the following

${\displaystyle \frac{(4\times {10}^7)(6\times {10}^{-5})}{8\times {10}^4}}$

Simplify the following

${\displaystyle \frac{4a{b}^2(-5a{b}^3)}{10a^2{b}^2}}$

Simplify the following

${\displaystyle {\left(\frac{x^2{y}^2}{a^2{b}^3}\right)}^n}$

Simplify the following

${\displaystyle \frac{{\left(a^{3n-9}\right)}^6}{a^{2n-4}}}$

If a = 3 and b = -2, find the values of :

a^a + b^b

If a = 3 and b = -2, find the values of :

a^b + b^a

If a = 3 and b = -2, find the values of :

(a + b)^ab

Prove that:

${\displaystyle {\left(\frac{x^a}{x^b}\right)}^{a^2+ab+b^2}\times {\left(\frac{x^b}{x^c}\right)}^{b^2+bc+c^2}\times {\left(\frac{x^c}{x^a}\right)}^{c^2+ca+a^2}=1}$

Prove that:

${\displaystyle {\left(\frac{x^a}{x^b}\right)}^c\times {\left(\frac{x^b}{x^c}\right)}^a\times {\left(\frac{x^c}{x^a}\right)}^b=1}$

Prove that:

${\displaystyle \frac{1}{1+x^{a-b}}+\frac{1}{1+x^{b-a}}=1}$

Prove that:

${\displaystyle \frac{1}{1+x^{b-a}+x^{c-a}}+\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{b-c}+x^{a-c}}=1}$

Prove that:

${\displaystyle \frac{a+b+c}{a^{-1}{b}^{-1}+b^{-1}{c}^{-1}+c^{-1}{a}^{-1}}=abc}$

Prove that:

${\displaystyle {(a^{-1}+b^{-1})}^{-1}=\frac{ab}{a+b}}$

If abc = 1, show that ${\displaystyle \frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}=1}$

Simplify the following:

${\displaystyle \frac{3^n\times 9^{n+1}}{3^{n-1}\times 9^{n-1}}}$

Simplify the following:

${\displaystyle \frac{5\times {25}^{n+1}-25\times 5^{2n}}{5\times 5^{2n+3}-{25}^{n+1}}}$

Simplify the following:

${\displaystyle \frac{5^{n+3}-6\times 5^{n+1}}{9\times 5^x-2^2\times 5^n}}$

Simplify the following:

${\displaystyle \frac{6{\left(8\right)}^{n+1}+16{\left(2\right)}^{3n-2}}{10{\left(2\right)}^{3n+1}-7{\left(8\right)}^n}}$

Solve the following equation for x:

${\displaystyle 7^{2x+3}=1}$

Solve the following equation for x:

${\displaystyle 2^{x+1}=4^{x-3}}$

Solve the following equation for x:

${\displaystyle 2^{5x+3}=8^{x+3}}$

Solve the following equation for x:

${\displaystyle 4^{2x}=\frac{1}{32}}$

Solve the following equation for x:

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

Solve the following equation for x:

${\displaystyle 2^{3x-7}=256}$

Solve the following equations for x:

${\displaystyle 2^{2x}-2^{x+3}+2^4=0}$

Solve the following equations for x:

${\displaystyle 3^{2x+4}+1={2.3}^{x+2}}$

If 49392 = a⁴b²c³, find the values of a, b and c, where a, b and c are different positive primes.

If ${\displaystyle 1176=2^a{3}^b{7}^c,}$ find a, b and c.

Given ${\displaystyle 4725=3^a{5}^b{7}^c,}$ find

(i) the integral values of a, b and c

(ii) the value of ${\displaystyle 2^{-a}{3}^b{7}^c}$

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

Assuming that x, y, z are positive real numbers, simplify the following:

${\displaystyle {\left(\sqrt{x^{-3}}\right)}^5}$

Assuming that x, y, z are positive real numbers, simplify the following:

${\displaystyle \sqrt{x^3{y}^{-2}}}$

Assuming that x, y, z are positive real numbers, simplify the following:

${\displaystyle {\left(x^{\frac{-2}{3}}{y}^{\frac{-1}{2}}\right)}^2}$

Assuming that x, y, z are positive real numbers, simplify the following:

${\displaystyle {\left(\sqrt{x}\right)}^{\frac{-2}{3}}\sqrt{y^4}÷\sqrt{x{y}^{\frac{-1}{2}}}}$

Assuming that x, y, z are positive real numbers, simplify the following:

${\displaystyle \sqrt[5]{243x^{10}{y}^5{z}^{10}}}$

Assuming that x, y, z are positive real numbers, simplify the following:

${\displaystyle {\left(\frac{x^{-4}}{y^{-10}}\right)}^{\frac{5}{4}}}$

Assuming that x, y, z are positive real numbers, simplify the following:

${\displaystyle {\left(\frac{\sqrt{2}}{\sqrt{3}}\right)}^5{\left(\frac{6}{7}\right)}^2}$

Simplify:

${\displaystyle {\left({16}^{-\frac{1}{5}}\right)}^{\frac{5}{2}}}$

Simplify:

${\displaystyle \sqrt[5]{{\left(32\right)}^{-3}}}$

Simplify:

${\displaystyle \sqrt[3]{{\left(343\right)}^{-2}}}$

Simplify:

${\displaystyle {\left(0.001\right)}^{\frac{1}{3}}}$

Simplify:

${\displaystyle \frac{{\left(25\right)}^{\frac{3}{2}}\times {\left(243\right)}^{\frac{3}{5}}}{{\left(16\right)}^{\frac{5}{4}}\times {\left(8\right)}^{\frac{4}{3}}}}$

Simplify:

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

Simplify:

${\displaystyle {\left(\frac{5^{-1}\times 7^2}{5^2\times 7^{-4}}\right)}^{\frac{7}{2}}\times {\left(\frac{5^{-2}\times 7^3}{5^3\times 7^{-5}}\right)}^{-\frac{5}{2}}}$

Prove that:

${\displaystyle \sqrt{3\times 5^{-3}}÷\sqrt[3]{3^{-1}}\sqrt{5}\times \sqrt[6]{3\times 5^6}=\frac{3}{5}}$

Prove that:

${\displaystyle 9^{\frac{3}{2}}-3\times 5^0-{\left(\frac{1}{81}\right)}^{-\frac{1}{2}}=15}$

Prove that:

${\displaystyle {\left(\frac{1}{4}\right)}^{-2}-3\times 8^{\frac{2}{3}}\times 4^0+{\left(\frac{9}{16}\right)}^{-\frac{1}{2}}=\frac{16}{3}}$

Prove that:

${\displaystyle \frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}}\times 4^{\frac{1}{4}}}{{10}^{-\frac{1}{5}}\times 5^{\frac{3}{5}}}÷\frac{3^{\frac{4}{3}}\times 5^{-\frac{7}{5}}}{4^{-\frac{3}{5}}\times 6}=10}$

Prove that:

${\displaystyle \sqrt{\frac{1}{4}}+{\left(0.01\right)}^{-\frac{1}{2}}-{\left(27\right)}^{\frac{2}{3}}=\frac{3}{2}}$

Prove that:

${\displaystyle \frac{2^n+2^{n-1}}{2^{n+1}-2^n}=\frac{3}{2}}$

Prove that:

${\displaystyle {\left(\frac{64}{125}\right)}^{-\frac{2}{3}}+\frac{1}{{\left(\frac{256}{625}\right)}^{\frac{1}{4}}}+\left(\frac{\sqrt{25}}{\sqrt[3]{64}}\right)=\frac{65}{16}}$

Prove that:

${\displaystyle \frac{3^{-3}\times 6^2\times \sqrt{98}}{5^2\times \sqrt[3]{\frac{1}{25}}\times {\left(15\right)}^{-\frac{4}{3}}\times 3^{\frac{1}{3}}}=28\sqrt{2}}$

Prove that:

${\displaystyle \frac{{\left(0.6\right)}^0-{\left(0.1\right)}^{-1}}{{\left(\frac{3}{8}\right)}^{-1}{\left(\frac{3}{2}\right)}^3+{\left(\frac{-1}{3}\right)}^{-1}}=\frac{-3}{2}}$

Show that:

${\displaystyle \frac{1}{1+x^{a-b}}+\frac{1}{1+x^{b-a}}=1}$

Show that:

${\displaystyle {[\left\{\frac{x^{a(a-b)}}{x^{a(a+b)}}\right\}÷\left\{\frac{x^{b(b-a)}}{x^{b(b+a)}}\right\}]}^{a+b}=1}$

Show that:

${\displaystyle {\left(x^{\frac{1}{a-b}}\right)}^{\frac{1}{a-c}}{\left(x^{\frac{1}{b-c}}\right)}^{\frac{1}{b-a}}{\left(x^{\frac{1}{c-a}}\right)}^{\frac{1}{c-b}}=1}$

Show that:

${\displaystyle {\left(\frac{x^{a^2+b^2}}{x^{ab}}\right)}^{a+b}{\left(\frac{x^{b^2+c^2}}{x^{bc}}\right)}^{b+c}{\left(\frac{x^{c^2+a^2}}{x^{ac}}\right)}^{a+c}=x^{2(a^3+b^3+c^3)}}$

Show that:

${\displaystyle {\left(x^{a-b}\right)}^{a+b}{\left(x^{b-c}\right)}^{b+c}{\left(x^{c-a}\right)}^{c+a}=1}$

Show that:

${\displaystyle {\left\{{\left(x^{a-a^{-1}}\right)}^{\frac{1}{a-1}}\right\}}^{\frac{a}{a+1}}=x}$

Show that:

${\displaystyle {\left(\frac{a^{x+1}}{a^{y+1}}\right)}^{x+y}{\left(\frac{a^{y+2}}{a^{z+2}}\right)}^{y+z}{\left(\frac{a^{z+3}}{a^{x+3}}\right)}^{z+x}=1}$

Show that:

${\displaystyle {\left(\frac{3^a}{3^b}\right)}^{a+b}{\left(\frac{3^b}{3^c}\right)}^{b+c}{\left(\frac{3^c}{3^a}\right)}^{c+a}=1}$

If 2^x = 3^y = 12^z, show that ${\displaystyle \frac{1}{z}=\frac{1}{y}+\frac{2}{x}}$

If 2^x = 3^y = 6^-z, show that ${\displaystyle \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0}$

If a^x = b^y = c^z and b² = ac, show that ${\displaystyle y=\frac{2zx}{z+x}}$

If 3^x = 5^y = (75)^z, show that ${\displaystyle z=\frac{xy}{2x+y}}$

If ${\displaystyle {27}^x=\frac{9}{3^x},}$ find x.

Find the value of x in the following:

${\displaystyle 2^{5x}÷2x=\sqrt[5]{2^{20}}}$

Find the value of x in the following:

${\displaystyle {\left(2^3\right)}^4={\left(2^2\right)}^x}$

Find the value of x in the following:

${\displaystyle {\left(\frac{3}{5}\right)}^x{\left(\frac{5}{3}\right)}^{2x}=\frac{125}{27}}$

Find the value of x in the following:

${\displaystyle 5^{x-2}\times 3^{2x-3}=135}$

Find the value of x in the following:

${\displaystyle 2^{x-7}\times 5^{x-4}=1250}$

Find the value of x in the following:

${\displaystyle {\left(\sqrt[3]{4}\right)}^{2x+\frac{1}{2}}=\frac{1}{32}}$

Find the value of x in the following:

${\displaystyle 5^{2x+3}=1}$

Find the value of x in the following:

${\displaystyle {\left(13\right)}^{\sqrt{x}}=4^4-3^4-6}$

Find the value of x in the following:

${\displaystyle {\left(\sqrt{\frac{3}{5}}\right)}^{x+1}=\frac{125}{27}}$

If ${\displaystyle x=2^{\frac{1}{3}}+2^{\frac{2}{3}},}$ Show that x³ - 6x = 6

Determine ${\displaystyle {\left(8x\right)}^x,}$If ${\displaystyle 9^{x+2}=240+9^x}$

If ${\displaystyle 3^{x+1}=9^{x-2},}$ find the value of ${\displaystyle 2^{1+x}}$

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

If ${\displaystyle 5^{3x}=125}$ and ${\displaystyle {10}^y=0.001,}$ find x and y.

Solve the following equation:

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

Solve the following equation:

${\displaystyle 4^{2x}={\left(\sqrt[3]{16}\right)}^{-\frac{6}{y}}={\left(\sqrt{8}\right)}^2}$

Solve the following equation:

${\displaystyle 3^{x-1}\times 5^{2y-3}=225}$

Solve the following equation:

${\displaystyle 8^{x+1}={16}^{y+2}}$ and, ${\displaystyle {\left(\frac{1}{2}\right)}^{3+x}={\left(\frac{1}{4}\right)}^{3y}}$

Solve the following equation:

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

Solve the following equation:

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

If a and b are distinct primes such that ${\displaystyle \sqrt[3]{a^6{b}^{-4}}=a^x{b}^{2y},}$ find x and y.

If a and b are different positive primes such that

${\displaystyle {\left(\frac{a^{-1}{b}^2}{a^2{b}^{-4}}\right)}^7÷\left(\frac{a^3{b}^{-5}}{a^{-2}{b}^3}\right)=a^x{b}^y,}$ find x and y.

If a and b are different positive primes such that

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

If ${\displaystyle 2^x\times 3^y\times 5^z=2160,}$ find x, y and z. Hence, compute the value of ${\displaystyle 3^x\times 2^{-y}\times 5^{-z}.}$

If 1176 = ${\displaystyle 2^a\times 3^b\times 7^c,}$ find the values of a, b and c. Hence, compute the value of ${\displaystyle 2^a\times 3^b\times 7^{-c}}$ as a fraction.

Simplify:

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

Simplify:

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

Show that:

${\displaystyle \frac{{(a+\frac{1}{b})}^m\times {(a-\frac{1}{b})}^n}{{(b+\frac{1}{a})}^m\times {(b-\frac{1}{a})}^n}={\left(\frac{a}{b}\right)}^{m+n}}$

If ${\displaystyle a=x^{m+n}{y}^l,b=x^{n+l}{y}^m}$ and ${\displaystyle c=x^{l+m}{y}^n,}$ Prove that ${\displaystyle a^{m-n}{b}^{n-l}{c}^{l-m}=1}$

If ${\displaystyle x=a^{m+n},}$ ${\displaystyle y=a^{n+l}}$ and ${\displaystyle z=a^{l+m},}$ prove that ${\displaystyle x^m{y}^n{z}^l=x^n{y}^l{z}^m}$

Write $${\left(625\right)}^{-1/4}$$ in decimal form.

If 2⁴ × 4² =16^x, then find the value of x.

If 3^x-1 = 9 and 4^y+2 = 64, what is the value  of $$\frac{x}{y}$$ ?

Write the value of  $$\sqrt[3]{7}\times \sqrt[3]{49}.$$

Write $${\left(\frac{1}{9}\right)}^{-1/2}\times (64{)}^{-1/3}$$ as a rational number.

Write the value of $$\sqrt[3]{125\times 27}$$.

For any positive real number x, find the value of $${\left(\frac{x^a}{x^b}\right)}^{a+b}\times {\left(\frac{x^b}{x^c}\right)}^{b+c}\times {\left(\frac{x^c}{x^a}\right)}^{c+a}$$.

Write the value of $${\{5(8^{1/3}+{27}^{1/3}{)}^3\}}^{1/4}.$$

Simplify $${\left[{\left\{{\left(625\right)}^{-1/2}\right\}}^{-1/4}\right]}^2$$

For any positive real number x, write the value of  $${\left\{{\left(x^a\right)}^b\right\}}^{\frac{1}{ab}}{\left\{{\left(x^b\right)}^c\right\}}^{\frac{1}{bc}}{\left\{{\left(x^c\right)}^a\right\}}^{\frac{1}{ca}}$$

If (x − 1)³ = 8, What is the value of (x + 1)² ?

The value of $${\{2-3(2-3{)}^3\}}^3$$ is 

The value of x − y^x-y when x = 2 and y = −2 is

The product of the square root of x with the cube root of x is

The seventh root of x divided by the eighth root of x is

The square root of 64 divided by the cube root of 64 is

Which of the following is (are) not equal to $${\left\{{\left(\frac{5}{6}\right)}^{1/5}\right\}}^{-1/6}$$ ?

When simplified $$(x^{-1}+y^{-1}{)}^{-1}$$ is equal to

If $$8^{x+1}$$ = 64 , what is the value of $$3^{2x+1}$$ ?

If (2³)² = 4^x, then 3^x =

If x^{-2 =} 64, then x^1/3+x⁰ =

When simplified $${(-\frac{1}{27})}^{-2/3}$$ is 

Which one of the following is not equal to $${\left(\sqrt[3]{8}\right)}^{-1/2}?$$

Which one of the following is not equal to $${\left(\frac{100}{9}\right)}^{-3/2}$$?

If a, b, c are positive real numbers, then  $$\sqrt{a^{-1}b}\times \sqrt{b^{-1}c}\times \sqrt{c^{-1}a}$$ is equal to

 ${\displaystyle {\left(\frac{2}{3}\right)}^x{\left(\frac{3}{2}\right)}^{2x}=\frac{81}{16}}$then x = 

The value of $${\{8^{-4/3}÷2^{-2}\}}^{1/2}$$ is

If a, b, c are positive real numbers, then  $$\sqrt[5]{3125a^{10}{b}^5{c}^{10}}$$  is equal to

If a, m, n are positive ingegers, then $${\left\{\sqrt[m]{\sqrt[n]{a}}\right\}}^{mn}$$ is equal to

If x = 2 and y = 4, then $${\left(\frac{x}{y}\right)}^{x-y}+{\left(\frac{y}{x}\right)}^{y-x}=$$

The value of m for which $${\left[{\left\{{\left(\frac{1}{7^2}\right)}^{-2}\right\}}^{-1/3}\right]}^{1/4}=7^m,$$ is

The value of $${\{{(23+2^2)}^{2/3}+(140-19{)}^{1/2}\}}^2,$$ is 

(256)^0.16 × (256)^0.09

If 10^2y = 25, then 10^-y equals

If 9^x+2 = 240 + 9^x, then x =

If x is a positive real number and x² = 2, then x³ =

If $$\frac{x}{x^{1.5}}=8x^{-1}$$ and x > 0, then x =

If g = ${\displaystyle t^{\frac{2}{3}}+4t^{-\frac{1}{2}}}$, what is the value of g when t = 64?

If $$4x-4x^{-1}=24,$$ then (2x)^x equals

When simplified $$(256){}^{-(4^{-3/2})}$$ is

If $$\frac{3^{2x-8}}{225}=\frac{5^3}{5^x},$$  then x =

The value of 64^-1/3 (64^1/3-64^2/3), is

If $$\sqrt{5^n}=125$$ then  ${\displaystyle 5n\sqrt{64}}$=

If (16)^2x+3 =(64)^x+3, then 4^2x-2 =

If $$2^{-m}\times \frac{1}{2^m}=\frac{1}{4},$$ then $$\frac{1}{14}\{(4^m{)}^{1/2}+{\left(\frac{1}{5^m}\right)}^{-1}\}$$  is equal to

If $$\frac{2^{m+n}}{2^{n-m}}=16$$, $$\frac{3^p}{3^n}=81$$ and $$a=2^{1/10}$$,than  $$\frac{a^{2m+n-p}}{(a^{m-2n+2p}{)}^{-1}}=$$

If $$\frac{3^{5x}\times {81}^2\times 6561}{3^{2x}}=3^7$$  then x =

If o <y <x, which statement must be true?

If 10^x = 64, what is the value of $${10}^{\frac{x}{2}}+1?$$

$$\frac{5^{n+2}-6\times 5^{n+1}}{13\times 5^n-2\times 5^{n+1}}$$  is equal to

If $$\sqrt{2^n}=1024,$$ then $${3^2}^{(\frac{n}{4}-4)}=$$