Miscellaneous Examples Using the Laws of Exponents

Find $m$ so that ${\left(\frac{2}{9}\right)}^3\times {\left(\frac{2}{9}\right)}^6={\left(\frac{2}{9}\right)}^{2m-1}$

By what number should $(-4{)}^5$ be divided so that the quotient may be equal to $(-4{)}^3$ ?

List two ways to express $4^5$ as a product of powers.

Explain why the exponents cannot be added in the product ${14}^3\times {18}^3$.

Compare $3\times 2$,$3^2$ and $2^3$.

Explain the difference between $(-5{)}^2$ and $-5^2$.

Find the reciprocal of the rational number ${\left(\frac{1}{2}\right)}^2÷{\left(\frac{2}{3}\right)}^3$

If $\frac{\mathrm{p}}{\mathrm{q}}={\left(\frac{3}{2}\right)}^2÷{\left(\frac{9}{4}\right)}^0$, find the value of ${\left(\frac{\mathrm{p}}{\mathrm{q}}\right)}^3$.

Express the given information in Scientific notation and then arrange them in descending order of their size.

$\begin{array}{|cll|}\hline \text{ Sl.No. }& \text{ Name of the Planet }& \text{ Mass (in kg) }\\ 1.& \text{ Mercury }& 330000000000000000000000\\ 2.& \text{ Venus }& 4870000000000000000000000\\ 3.& \text{ Earth }& 5980000000000000000000000\\ 4.& \text{ Mars }& 642000000000000000000000\\ 5.& \text{ Jupiter }& 1900000000000000000000000000\\ 6.& \text{ Saturn }& 569000000000000000000000000\\ 7.& \text{ Uranus }& 86900000000000000000000000\\ 8.& \text{ Neptune }& 102000000000000000000000000\\ 9.& \text{ Pluto }& 13100000000000000000000\\ \hline\end{array}$

Express the given information in Scientific notation (standard form) and then arrange them in ascending order of their size.

$\begin{array}{|clc|}\hline \text{ S1.No. }& \text{ Deserts of the World }& \text{ Area (Sq. Kilometres) }\\ 1.& \text{ Kalahar1, South Africa }& 932,400\\ 2.& \text{ Thar, India }& 199,430\\ 3.& \text{ Gibson, Australia }& 155,400\\ 4.& \text{ Great Victorla, Australia }& 647,500\\ 5.& \text{ Sahara, North Africa }& 8,598,800\\ \hline\end{array}$

${\left(\frac{2}{5}\right)}^3÷{\left(\frac{5}{2}\right)}^3=1$

${\left(\frac{4}{3}\right)}^5\times {\left(\frac{5}{7}\right)}^5={\left(\frac{4}{3}+\frac{5}{7}\right)}^5$

${\left(\frac{5}{8}\right)}^9÷{\left(\frac{5}{8}\right)}^4={\left(\frac{5}{8}\right)}^4$

${\left(\frac{7}{3}\right)}^2\times {\left(\frac{7}{3}\right)}^5={\left(\frac{7}{3}\right)}^{10}$

$5^0\times {25}^0\times {125}^0={\left(5^0\right)}^6$

Arrange in descending order :

$2^{2+3} , {\left(2^2\right)}^3 , 2\times 2^2 , \frac{3^5}{3^2} , 3^2\times 3^0 , 2^3\times 5^2$

Arrange in ascending order :

$2^5 , 3^3 , 2^3\times 2 , {\left(3^3\right)}^2 , 3^5 , 4^0 , 2^3\times 3^1$

$4^0+5^0+6^0=(4+5+6{)}^0$

$8\times {10}^6+2\times {10}^4+5\times {10}^2+9\times {10}^0=8020509$

$4\times {10}^5+3\times {10}^4+2\times {10}^3+1\times {10}^0=432010$