Laws of Exponents for Real Numbers

Q.

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

Q.

If $2^x=3^y={12}^z,Showthat\frac{1}{z}=\frac{1}{y}+\frac{2}{x}$.

Q.

Prove that:

(i) $\frac{1}{1+x^{a-b}}+\frac{1}{1+x^{b-a}}=1$

(ii) $\frac{1}{1+x^{b-a}+x^{c-a}}+\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{b-c}+x^{a-c}}=1$

Q.

If ${10}^x=64,whatisthevalueof{10}^{\frac{x}{2}+1}?$

1. 80

2. 81

3. 18

4. 42

Q.

Solve the following equations for x:

(i) $2^{2x}-2^{x+3}+2^4$ =0

(ii) $3^{2x+4}+1=2.3^{x+2}$

Q.

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

Q.

$\frac{5^{n+2}-6\times 5^{n+1}}{13\times 5^n-2\times 5^{n+1}}isequalto$

Q.

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

1. 2

2. 4

3. 3

4. 1

Q.

If x + y + z = 0, show that $x^3$ + $y^3$ + $z^3$ = 3xyz.

Q.

If $x^{-2}=64,then{x}^{\frac{1}{3}}+x^0=$

1. $\frac{3}{2}$

2. $\frac{2}{3}$

3. 2

4. 3

Q.

$Determine(8x{)}^x,if{9}^{x+2}=240+9^x.$

Q.

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

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

(ii) $\sqrt{x^3{y}^{-2}}$

(iii) $(x^{-2/3}{y}^{-1/2}{)}^2$

(iv) $(\sqrt{x}{)}^{-2/3}\sqrt{y^4}÷\sqrt{x{y}^{-1/2}}$

(v) $5\sqrt{243x^{10}{y}^5{z}^{10}}$

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

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

Q.

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

1. $5\sqrt{5}$

2. $\sqrt{5}$

3. $25\sqrt{5}$

4. 125

Q.

Prove that :

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

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

Q.

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

1. 625

2. $\frac{1}{5}$

3. 25

4. $\frac{1}{125}$

Q.

Solve the following equations for x:

(i) $7^{2x+3}=1$

(ii) $2^{x+1}=4^{x-3}$

(iii)$2^{5x+3}=8^{x+3}$

(iv) $4^{2x}=\frac{1}{32}$

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

(vi)$2^{3x-7}=256$

Q.

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

(i) $a^a+b^b\left(ii\right)a^b+b^a\left(iii\right)(a+b{)}^{ab}$

Q.

If$x=2^{\frac{1}{3}}+2^{\frac{2}{3}},Showthat{x}^3-6x=6.$

Q.

The simplified form of ${(-\frac{1}{27})}^{\frac{-2}{3}}$ is

Q.

Evaluate ${\left(\frac{81}{49}\right)}^{-\frac{3}{2}}$

Q.

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

Q.

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

Q.

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

1. 1

2. 3

3. 9

4. 27

Q.

What is the unit digit in $7^{105}$?

Q.

The number of consecutive zeros in $2^3\times 3^4\times 5^4\times 7$, is

1. 4

2. 5

3. 2

4. 3

Q.

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

Q.

If x is a positive real number and $x^2=2,then{x}^3=$

1. $\sqrt{2}$

2. $3\sqrt{2}$

3. $2\sqrt{2}$

4. 4

Q.

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

Q.

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

Q.

Simplify: $\left(i\right){\left(\frac{x^{a+b}}{x^c}\right)}^{a-b}{\left(\frac{x^{b+c}}{x^a}\right)}^{b-c}{\left(\frac{x^{c+a}}{x^b}\right)}^{c-a}\left(ii\right)\sqrt[lm]{lm\frac{x^l}{x^m}}\times \sqrt[mn]{mn\frac{x^m}{x^n}}\times \sqrt[nl]{nl\frac{x^n}{x^l}}$