Free Exponents and Powers 02 Practice Test - 8th Grade 

We know that $a^{-m}=\frac{1}{a^m}$ for any non-zero integer $a$ and integer $m.$
$⟹7^{-3}=\frac{1}{7^3}$
                 $=\frac{1}{7\times 7\times 7}=\frac{1}{343}$

Multiplicative inverse of a non-zero $x$ is $\frac{1}{x}$; so that when these two are multiplied, the product value is 1. Therefore, the multiplicative inverse of ${10}^{-7}$should be a number which when multiplied with ${10}^{-7}$ gives 1.
Let the multiplicative inverse of ${10}^{-7}$ be $x.$
Then, ${10}^{-7}\times x=1.$
$\Rightarrow x=\frac{1}{{10}^{-7}}={10}^7$
$(\because a^{-n}=\frac{1}{a^n}⟹\frac{1}{a^{-n}}=\frac{1}{\frac{1}{a^n}}=a^n)$
Thus, the multiplicative inverse of ${10}^{-7}$ is ${10}^7.$

Given expression is

$4^3\times 4^{-2}\times {16}^2=4^3\times 4^{-2}\times {\left(\right(4{)}^2)}^2=4^3\times 4^{-2}\times 4^4(\because (a^m{)}^n=a^{mn})=4^{3-2+4}(\because a^m\times a^n=a^{m+n})=4^5$ 

$\because a^{-n}=\frac{1}{a^n}fora\ne 0$
$⟹(\frac{1}{5}{)}^{-3}=\frac{1}{(\frac{1}{5}{)}^3}=(\frac{5}{1}{)}^3=5^3=125$
Similarly, $(\frac{1}{3}{)}^{-4}=\frac{1}{(\frac{1}{3}{)}^4}=(\frac{3}{1}{)}^4=3^4=81$ and 
$(\frac{1}{4}{)}^{-3}=\frac{1}{(\frac{1}{4}{)}^3}=(\frac{4}{1}{)}^3=4^3=64$

$⟹\frac{(\frac{1}{5}{)}^{-3}-(\frac{1}{3}{)}^{-4}}{(\frac{1}{4}{)}^{-3}}=\frac{125-81}{64}=\frac{44}{64}=\frac{11}{16}$

$1,540,000,000=1.54\times 1000000000=1.54\times {10}^9$

The number written at the bottom is the base whereas the number at the superscript is known as the exponent. In the given example, 5 is the base and 3 is the exponent. 

3676.48 can be written in expanded form as:
$3\times {10}^3+6\times {10}^2+7\times {10}^1+6\times {10}^0+4\times {10}^{-1}+8\times {10}^{-2}$. 
 $a=3,b=6,c=1,d=-1,e=8a+b-c-d-e=3+6-1+1-8=1$

Given, $(2{)}^{m+1}\times 2^5=4^{-2}$
$\Rightarrow$ $(2{)}^{m+1}\times 2^5={\left(\right(2{)}^2)}^{-2}$
$\Rightarrow$ $(2{)}^{m+1+5}=(2{)}^{-4}$ 
Since the bases on  both the sides  are equal, therefore the exponents must also be equal.  This means, $m+1+5=-4$, or $m=-10$.

$5.06\times {10}^{-4}=\frac{5.06}{10000}=0.000506$

The total distance travelled by the spaceship between Earth and Saturn = Distance between (Earth & Mars + Mars & Saturn)
               2.2253x${10}^8$  kms + 1.19666 x ${10}^{9}kms$   
              = 2.2253x${10}^8$  kms + 11.9666 x ${10}^{8}kms$
                =14.1919 x${10}^8$ kms
                = 1.41919x${10}^9$$\times {10}^{3}m$ 
                = 1.41919x ${10}^{12}m=1419190000000m$