Examples (Revised) - Chapter 12 - Exponents & Powers - Ncert Solutions class 8 - Maths

You can Download the Examples (Revised) - Chapter 12 - Exponents & Powers - Ncert Solutions class 8 - Maths with expert answers for all chapters. Perfect for Tamil & English Medium students to revise the syllabus and score more marks in board exams. Download and share it with your friends

Share this to Friend on WhatsApp

Chapter 10: Exponents & Powers - NCERT Solutions for Class 8 Maths

Example 1:

Find the value of
(i) $2^{-3}$
(ii) $\frac{1}{3^{-2}}$

Solution:
(i) $2^{-3}=\frac{1}{2^3}=\frac{1}{8}$
(ii) $\frac{1}{3^{-2}}=3^2=3 \times 3=9$

Example 2:

Simplify
(i) $(-4)^5 \times(-4)^{-10}$
(ii) $2^5 \div 2^{-6}$

Solution:
(i) $(-4)^5 \times(-4)^{-10}=(-4)^{(5-10)}=(-4)^{-5}=\frac{1}{(-4)^5} \quad\left(a^m \times a^n=a^{m+n}, a^{-m}=\frac{1}{a^m}\right)$
(ii) $2^5 \div 2^{-6}=2^{5-(-6)}=2^{11} \quad\left(a^m \div a^n=a^{m-n}\right)$

Example 3:

Express $4^{-3}$ as a power with the base 2.
Solution:

We have, $4=2 \times 2=2^2$
Therefore, $(4)^{-3}=(2 \times 2)^{-3}=\left(2^2\right)^{-3}=2^{2 \times(-3)}=2^{-6} \quad\left[\left(a^m\right)^n=a^{m n}\right]$
Example 4:

Simplify and write the answer in the exponential form.
(i) $\left(2^5 \div 2^8\right)^5 \times 2^{-5}$
(ii) $(-4)^{-3} \times(5)^{-3} \times(-5)^{-3}$
(iii) $\frac{1}{8} \times(3)^{-3}$
(iv) $(-3)^4 \times\left(\frac{5}{3}\right)^4$

Solution:
(i) $\left(2^5 \div 2^8\right)^5 \times 2^{-5}=\left(2^{5-8}\right)^5 \times 2^{-5}=\left(2^{-3}\right)^5 \times 2^{-5}=2^{-15-5}=2^{-20}=\frac{1}{2^{20}}$
(ii) $(-4)^{-3} \times(5)^{-3} \times(-5)^{-3}=[(-4) \times 5 \times(-5)]^{-3}=[100]^{-3}=\frac{1}{100^3}$
[using the law $a^m \times b^m=(a b)^m, a^{-m}=\frac{1}{a^m}$ ]
(iii) $\frac{1}{8} \times(3)^{-3}=\frac{1}{2^3} \times(3)^{-3}=2^{-3} \times 3^{-3}=(2 \times 3)^{-3}=6^{-3}=\frac{1}{6^3}$

(iv)
$
\begin{aligned}
(-3)^4 \times\left(\frac{5}{3}\right)^4 & =(-1 \times 3)^4 \times \frac{5^4}{3^4}=(-1)^4 \times 3^4 \times \frac{5^4}{3^4} \\
& =(-1)^4 \times 5^4=5^4 \quad\left[(-1)^4=1\right]
\end{aligned}
$

Example 5:

Find $m$ so that $(-3)^{m+1} \times(-3)^5=(-3)^7$
Solution:
$
\begin{aligned}
(-3)^{m+1} \times(-3)^5 & =(-3)^7 \\
(-3)^{m+1+5} & =(-3)^7 \\
(-3)^{m+6} & =(-3)^7
\end{aligned}
$

On both the sides powers have the same base different from 1 and -1 , so their exponents must be equal.

Therefore,
$
\begin{array}{r}
m+6=7 \\
m=7-6=1
\end{array}
$

$a^n=1$ only if $n=0$. This will work for any $a$. For $a=1,1^1=1^2=1^3=1^{-2}=\ldots=1$ or $(1)^n=$ 1 for infinitely many $n$.
For $a=-1$, $(-1)^0=(-1)^2=(-1)^4=(-1)^{-2}=\ldots=1$ or $(-1)^p=1$ for any even integer $p$.

Example 6:

Find the value of $\left(\frac{2}{3}\right)^{-2}$.
Solution:

$\left(\frac{2}{3}\right)^{-2}=\frac{2^{-2}}{3^{-2}}=\frac{3^2}{2^2}=\frac{9}{4}$

$
\left(\frac{2}{3}\right)^{-2}=\frac{2^{-2}}{3^{-2}}=\frac{3^2}{2^2}=\left(\frac{3}{2}\right)^2
$

In general, $\left(\frac{a}{b}\right)^{-m}=\left(\frac{b}{a}\right)^m$

Example 7:

Simplify (i) $\left\{\left(\frac{1}{3}\right)^{-2}-\left(\frac{1}{2}\right)^{-3}\right\} \div\left(\frac{1}{4}\right)^{-2}$
(ii) $\left(\frac{5}{8}\right)^{-7} \times\left(\frac{8}{5}\right)^{-5}$

Solution:
(i) $\left\{\left(\frac{1}{3}\right)^{-2}-\left(\frac{1}{2}\right)^{-3}\right\} \div\left(\frac{1}{4}\right)^{-2}=\left\{\frac{1^{-2}}{3^{-2}}-\frac{1^{-3}}{2^{-3}}\right\} \div \frac{1^{-2}}{4^{-2}}$

$
=\left\{\frac{3^2}{1^2}-\frac{2^3}{1^3}\right\} \div \frac{4^2}{1^2}=\{9-8\} \div 16=\frac{1}{16}
$
(ii)
$
\begin{aligned}
\left(\frac{5}{8}\right)^{-7} \times\left(\frac{8}{5}\right)^{-5} & =\frac{5^{-7}}{8^{-7}} \times \frac{8^{-5}}{5^{-5}}=\frac{5^{-7}}{5^{-5}} \times \frac{8^{-5}}{8^{-7}}=5^{(-7)-(-5)} \times 8^{(-5)-(-7)} \\
& =5^{-2} \times 8^2=\frac{8^2}{5^2}=\frac{64}{25}
\end{aligned}
$

Example 8:

Express the following numbers in standard form.
(i) 0.000035
(ii) 4050000

Solution:

(i) $0.000035=3.5 \times 10^{-5}$
(ii) $4050000=4.05 \times 10^6$

Example 9:

Express the following numbers in usual form.
(i) $3.52 \times 10^5$
(ii) $7.54 \times 10^{-4}$
(iii) $3 \times 10^{-5}$

Again we need to convert numbers in standard form into a numbers with the same exponents.

Solution:
(i) $3.52 \times 10^5=3.52 \times 100000=352000$
(ii) $7.54 \times 10^{-4}=\frac{7.54}{10^4}=\frac{7.54}{10000}=0.000754$
(iii) $3 \times 10^{-5}=\frac{3}{10^5}=\frac{3}{100000}=0.00003$