Q: Simplify: \$\displaystyle \frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}\$

\$ \cfrac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}\$\$=\cfrac { 3^{ -5 }\times (2\times 5)^{ -5 }\times 5^{ 3 } }{ 5^{ -7 }\times (2\times 3)^{ -5 } } \$\$=\cfrac { 3^{ -5 }\times 2^{ -5 }\times 5^{ -5 }\times 5^{ 3 } }{ 5^{ -7 }\times 2^{ -5 }\times 3^{ -5 } } \$\$=3^{ -5+5 }\times 2^{ -5+5 }\times 5^{ -5+3+7 }\$\$=5^{5}\$\$=3125\$

Q: Find the value of \$m\$ for which \$5^m \div 5^{-3} = 5^5\$

\$5^m \div 5^{-3} = 5^5\$\$\Rightarrow \displaystyle 5^m \div \frac{1}{5^3} = 5^5\$\$ \Rightarrow 5^m \times 5^3 = 5^5\$\$\Rightarrow 5^{m + 3} = 5^5\$\$\Rightarrow m + 3 = 5\$\$\Rightarrow m = 5 - 3 = 2\$

Q: Simplify and express the result in power notation with positive exponent.(i) \$(-4)^{5}\div (-4)^{8} \$(ii) \$\left(\dfrac{1}{2^{3}}\right)^{2} \$(iii) \$(-3)^{4}\times\left(\dfrac{5}{3}\right)^{4} \$(iv) \$\left(3^{-7}\div 3^{-10}\right)\times 3^{-5} \$(v) \$2^{-3}\times(-7)^{(-3)} \$

(i) \$(-4)^{5}\div (-4)^{8} = \dfrac{(-4)^5}{(-4)^8} = (-4)^{5-8} = (-4)^{-3} =\dfrac {1}{(-4)^3}\$(ii) \$\left (\dfrac{1}{2^{3}}\right )^{2} = \dfrac{1}{2^{3\times2}} =\dfrac{1}{2^{6}} \$(iii) \$(-3)^{4}\times \left (\dfrac{5}{3}\right )^{4} = (-1)^4\times(3)^4\times\left(\dfrac53\right)^4 = (-1)^{4}\times 5^{4} =5^{4} \$(iv) \$(3^{-7}\div 3^{-10})\times 3^{-5} = \left(\dfrac{3^{-7}}{3^{-10}}\right)\times(3)^{-5} = 3^{-7+10}\times 3^{-5} \$ \$ = 3^{3-5} \$\$ = 3^{-2} = \dfrac{1}{3^{2}} \$(v) \$2^{-3}\times(-7)^{(-3)} = \dfrac1{2^3}\times\dfrac1{(-7)^{3}} = \left (\dfrac{1}{2\times (-7)}\right )^{3} \$ \$ =\dfrac{1}{(-14)^{3}} \$

Q: \$a^{m}\times a^{n}\$ is equal to

Reffer image.So, \$a^{m}\$ x \$a^{n}=a^{(m+n)}\$

Q: Evaluate : \$(-4)^{-2}\$---

\$(-4)^{-2}=\dfrac{1}{(-4)^2}\$ \$\because a^{-m}=\dfrac{1}{a^m}\$\$=\dfrac{1}{-4*-4}\$\$=\dfrac1{16}\$\$\therefore(-4)^{-2}=\dfrac1{16}\$

Q: Find the value of \$2[(2^{-1}\times 4^{-1})\div 2^{-2}]\$.

Given, \$2[(2^{-1} \times 4^{-1}) \div 2^{-2})\$= \$2[(2^{-1} \times 4^{-1}) \div 2^{-2})\$= \$2[(\frac{1}{2} \times \frac{1}{4}) \div \frac{1}{4}]\$= \$2[(\frac{1}{2} \times \frac{1}{4}) \div \frac{1}{4}]\$= \$2[(\frac{1}{8}) \div \frac{1}{4}]\$= \$2[(\frac{1}{8}) \times \frac{4}{1}]\$= \$2(\frac{1}{2})\$= \$1\$

Q: Evaluate \$(5^{-1} \times 2^{-1}) \times 6^{-1}\$ is \$\frac{1}{m}\$Value of \$m\$ is

\$(5^{-1} \times 2^{-1}) \times 6^{-1}\$\$=\frac{1}{5}\times \frac{1}{2}\times \frac{1}{6}\$\$=\frac{1}{60}\$

Q: solve: \${ ({ 3 }^{ -1 } }+{ 4 }^{ -1 }+{ 5 }^{ -1 })^{ 0 }\$

\$\textbf{Use law of exponents}\$\$\text{Given,}\ (3^{-1}+4^{-1}+5^{-1})^0\$\$\text{Any non-zero number to the power 0 is 1}\$\$\therefore \textbf{The value of}\ \mathbf{(3^{-1}+4^{-1}+5^{-1})^0}\ \textbf{is 1}\$

Q: Express \$512\$ in exponential form

We can express \$512\$ as\$512=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\$\$512={2}^{9}\$

Q: Simplify:(i) \$(-4)^{5}\div (-4)^{8}\$(ii) \$\left (\dfrac {1}{2^{3}}\right )^{2}\$(iii) \$(-3)^{4} \times \left (\dfrac {5}{3}\right )^{4}\$(iv) \$\left (\dfrac {2}{3}\right )^{5}\times \left (\dfrac {3}{4}\right )^{2}\times \left (\dfrac {1}{5}\right )^{2}\$(v) \$(3^{-7} \div 3^{10})\times 3^{-5}\$(vi) \$\dfrac {2^{6}\times 3^{2} \times 2^{3} \times 3^{7}}{2^{8}\times 3^{6}}\$(vii) \$y^{a - b} \times y^{b- c}\times y^{c - a}\$

(i) \$(-4)^{5}\div (-4)^{8}\$\$\Rightarrow (-4)^{5}\times \dfrac{1}{(-4)^{8}}\$\$\Rightarrow (-1)\times (4)^{5}\times \dfrac{1}{(4)^{8}}\$\$\Rightarrow (-1)\times \dfrac{1}{(4)^{3}}\$\$\Rightarrow -\dfrac{1}{64}\$(ii) \$\left (\dfrac {1}{2^{3}}\right )^{2}\$\$\Rightarrow \left (\dfrac {1}{8}\right )^{2}\$\$\Rightarrow \dfrac {1}{64}\$(iii) \$(-3)^{4} \times \left (\dfrac {5}{3}\right )^{4}\$\$\Rightarrow (3)^{4} \times \left (\dfrac {5^4}{3^4}\right )\$\$\Rightarrow 5^4\$\$\Rightarrow 625\$(iv) \$\left (\dfrac {2}{3}\right )^{5}\times \left (\dfrac {3}{4}\right )^{2}\times \left (\dfrac {1}{5}\right )^{2}\$\$\Rightarrow \left (\dfrac {2^5}{3^5}\right )\times \left (\dfrac {3^2}{4^2}\right )\times \left (\dfrac {1}{5^2}\right )\$\$\Rightarrow \left (\dfrac {32}{3^3\times 3^2}\right )\times \left (\dfrac {3^2}{16}\right )\times \left (\dfrac {1}{25}\right )\$\$\Rightarrow \left (\dfrac {2}{3^3}\right )\times \left (\dfrac {1}{25}\right )\$\$\Rightarrow \left (\dfrac {2}{27}\right )\times \left (\dfrac {1}{25}\right )\$\$\Rightarrow \dfrac {2}{675}\$(v) \$(3^{-7} \div 3^{10})\times 3^{-5}\$\$\Rightarrow \left(\dfrac{1}{3^{7}} \div 3^{10}\right)\times \dfrac{1}{3^{5}}\$\$\Rightarrow \left(\dfrac{1}{3^{7}} \times \dfrac{1}{3^{10}}\right)\times \dfrac{1}{3^5}\$\$\Rightarrow \left(\dfrac{1}{3^{17}} \times \dfrac{1}{3^{5}}\right)\$\$\Rightarrow \dfrac{1}{3^{22}} \$(vi) \$\dfrac {2^{6}\times 3^{2} \times 2^{3} \times 3^{7}}{2^{8}\times 3^{6}}\$\$\Rightarrow \dfrac {2^{6}\times 3^{2} \times 2^{2}\times 2 \times 3^{6}\times 3}{2^{6}\times 2^2\times 3^{6}}\$\$\Rightarrow \dfrac {1\times 3^{2} \times 2^{2}\times 2 \times 3}{1\times1}\$\$\Rightarrow 216\$(vii) \$y^{a - b} \times y^{b- c}\times y^{c - a}\$\$\Rightarrow \dfrac{y^a}{ y^ b} \times \dfrac{y^b}{y^c}\times \dfrac{y^c}{ y^a}\$\$\Rightarrow 1\$Hence, this is the answer.

Question 1

Simplify: $\displaystyle \frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}$

Accepted Answer

$ \cfrac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}$$=\cfrac { 3^{ -5 }\times (2\times 5)^{ -5 }\times 5^{ 3 } }{ 5^{ -7 }\times (2\times 3)^{ -5 } } $$=\cfrac { 3^{ -5 }\times 2^{ -5 }\times 5^{ -5 }\times 5^{ 3 } }{ 5^{ -7 }\times 2^{ -5 }\times 3^{ -5 } } $$=3^{ -5+5 }\times 2^{ -5+5 }\times 5^{ -5+3+7 }$$=5^{5}$$=3125$

Question 2

Find the value of $m$ for which $5^m \div 5^{-3} = 5^5$

Accepted Answer

$5^m \div 5^{-3} = 5^5$$\Rightarrow \displaystyle 5^m \div \frac{1}{5^3} = 5^5$$ \Rightarrow 5^m \times 5^3 = 5^5$$\Rightarrow 5^{m + 3} = 5^5$$\Rightarrow m + 3 = 5$$\Rightarrow m = 5 - 3 = 2$

Question 3

Simplify and express the result in power notation with positive exponent.(i) $(-4)^{5}\div (-4)^{8} $(ii) $\left(\dfrac{1}{2^{3}}\right)^{2} $(iii) $(-3)^{4}\times\left(\dfrac{5}{3}\right)^{4} $(iv) $\left(3^{-7}\div 3^{-10}\right)\times 3^{-5} $(v) $2^{-3}\times(-7)^{(-3)} $

Accepted Answer

(i)    $(-4)^{5}\div (-4)^{8} = \dfrac{(-4)^5}{(-4)^8} = (-4)^{5-8} = (-4)^{-3} =\dfrac {1}{(-4)^3}$(ii)   $\left (\dfrac{1}{2^{3}}\right )^{2} = \dfrac{1}{2^{3\times2}} =\dfrac{1}{2^{6}} $(iii)   $(-3)^{4}\times \left (\dfrac{5}{3}\right )^{4} = (-1)^4\times(3)^4\times\left(\dfrac53\right)^4 =  (-1)^{4}\times 5^{4} =5^{4} $(iv)   $(3^{-7}\div 3^{-10})\times 3^{-5} = \left(\dfrac{3^{-7}}{3^{-10}}\right)\times(3)^{-5} = 3^{-7+10}\times 3^{-5} $ $ = 3^{3-5} $$ = 3^{-2} = \dfrac{1}{3^{2}} $(v)    $2^{-3}\times(-7)^{(-3)} = \dfrac1{2^3}\times\dfrac1{(-7)^{3}} = \left (\dfrac{1}{2\times (-7)}\right )^{3} $ $ =\dfrac{1}{(-14)^{3}} $

Question 4

$a^{m}\times a^{n}$ is equal to

Accepted Answer

Reffer image.So, $a^{m}$ x $a^{n}=a^{(m+n)}$

Question 5

Evaluate : $(-4)^{-2}$---

Accepted Answer

$(-4)^{-2}=\dfrac{1}{(-4)^2}$            $\because a^{-m}=\dfrac{1}{a^m}$$=\dfrac{1}{-4*-4}$$=\dfrac1{16}$$\therefore(-4)^{-2}=\dfrac1{16}$

Question 6

Find the value of $2[(2^{-1}\times 4^{-1})\div 2^{-2}]$.

Accepted Answer

Given, $2[(2^{-1} \times 4^{-1}) \div 2^{-2})$= $2[(2^{-1} \times 4^{-1}) \div 2^{-2})$= $2[(\frac{1}{2} \times \frac{1}{4}) \div \frac{1}{4}]$= $2[(\frac{1}{2} \times \frac{1}{4}) \div \frac{1}{4}]$= $2[(\frac{1}{8}) \div \frac{1}{4}]$= $2[(\frac{1}{8}) \times \frac{4}{1}]$= $2(\frac{1}{2})$= $1$

Question 7

Evaluate $(5^{-1} \times 2^{-1}) \times 6^{-1}$ is $\frac{1}{m}$Value of $m$ is

Accepted Answer

$(5^{-1} \times 2^{-1}) \times 6^{-1}$$=\frac{1}{5}\times \frac{1}{2}\times \frac{1}{6}$$=\frac{1}{60}$

Question 8

solve:        ${ ({ 3 }^{ -1 } }+{ 4 }^{ -1 }+{ 5 }^{ -1 })^{ 0 }$

Accepted Answer

$\textbf{Use law of exponents}$$\text{Given,}\ (3^{-1}+4^{-1}+5^{-1})^0$$\text{Any non-zero number to the power 0 is 1}$$\therefore \textbf{The value of}\ \mathbf{(3^{-1}+4^{-1}+5^{-1})^0}\ \textbf{is 1}$

Question 9

Express $512$ in exponential form

Accepted Answer

We can express $512$ as$512=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$$512={2}^{9}$

Question 10

Simplify:(i) $(-4)^{5}\div (-4)^{8}$(ii) $\left (\dfrac {1}{2^{3}}\right )^{2}$(iii) $(-3)^{4} \times \left (\dfrac {5}{3}\right )^{4}$(iv) $\left (\dfrac {2}{3}\right )^{5}\times \left (\dfrac {3}{4}\right )^{2}\times \left (\dfrac {1}{5}\right )^{2}$(v) $(3^{-7} \div 3^{10})\times 3^{-5}$(vi) $\dfrac {2^{6}\times 3^{2} \times 2^{3} \times 3^{7}}{2^{8}\times 3^{6}}$(vii) $y^{a - b} \times y^{b- c}\times y^{c - a}$

Accepted Answer

(i) $(-4)^{5}\div (-4)^{8}$$\Rightarrow (-4)^{5}\times \dfrac{1}{(-4)^{8}}$$\Rightarrow (-1)\times (4)^{5}\times \dfrac{1}{(4)^{8}}$$\Rightarrow (-1)\times \dfrac{1}{(4)^{3}}$$\Rightarrow -\dfrac{1}{64}$(ii) $\left (\dfrac {1}{2^{3}}\right )^{2}$$\Rightarrow \left (\dfrac {1}{8}\right )^{2}$$\Rightarrow \dfrac {1}{64}$(iii) $(-3)^{4} \times \left (\dfrac {5}{3}\right )^{4}$$\Rightarrow (3)^{4} \times \left (\dfrac {5^4}{3^4}\right )$$\Rightarrow 5^4$$\Rightarrow 625$(iv) $\left (\dfrac {2}{3}\right )^{5}\times \left (\dfrac {3}{4}\right )^{2}\times \left (\dfrac {1}{5}\right )^{2}$$\Rightarrow \left (\dfrac {2^5}{3^5}\right )\times \left (\dfrac {3^2}{4^2}\right )\times \left (\dfrac {1}{5^2}\right )$$\Rightarrow \left (\dfrac {32}{3^3\times 3^2}\right )\times \left (\dfrac {3^2}{16}\right )\times \left (\dfrac {1}{25}\right )$$\Rightarrow \left (\dfrac {2}{3^3}\right )\times \left (\dfrac {1}{25}\right )$$\Rightarrow \left (\dfrac {2}{27}\right )\times \left (\dfrac {1}{25}\right )$$\Rightarrow \dfrac {2}{675}$(v) $(3^{-7} \div 3^{10})\times 3^{-5}$$\Rightarrow \left(\dfrac{1}{3^{7}} \div 3^{10}\right)\times \dfrac{1}{3^{5}}$$\Rightarrow \left(\dfrac{1}{3^{7}} \times \dfrac{1}{3^{10}}\right)\times \dfrac{1}{3^5}$$\Rightarrow \left(\dfrac{1}{3^{17}} \times \dfrac{1}{3^{5}}\right)$$\Rightarrow \dfrac{1}{3^{22}} $(vi) $\dfrac {2^{6}\times 3^{2} \times 2^{3} \times 3^{7}}{2^{8}\times 3^{6}}$$\Rightarrow \dfrac {2^{6}\times 3^{2} \times 2^{2}\times 2 \times 3^{6}\times 3}{2^{6}\times 2^2\times 3^{6}}$$\Rightarrow \dfrac {1\times 3^{2} \times 2^{2}\times 2 \times 3}{1\times1}$$\Rightarrow 216$(vii) $y^{a - b} \times y^{b- c}\times y^{c - a}$$\Rightarrow \dfrac{y^a}{ y^ b} \times \dfrac{y^b}{y^c}\times \dfrac{y^c}{ y^a}$$\Rightarrow 1$Hence, this is the answer.