NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

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Updated at 2021-02-11

NCERT solutions for class 8 Maths Chapter 12 Exponents And Powers Exercise 12.1

1.Evaluate:
(i) ${3}^{?2}$

(ii) $\left(?4{\right)}^{?2}$

(iii) $\left(\frac{1}{2}{\right)}^{?5}$

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

(i) We have:
${3}^{?2}=\frac{1}{{3}^{2}}=\frac{1}{9}\phantom{\rule{1em}{0ex}}$[?${a}^{?n}=\frac{1}{{a}^{n}}$]

(ii)We have:
$\left(?4{\right)}^{?2}=\frac{1}{\left(?4{\right)}^{2}}=\frac{1}{16}\phantom{\rule{1em}{0ex}}$[?${a}^{?n}=\frac{1}{{a}^{n}}$]

(iii)We have:
$\left(\frac{1}{2}{\right)}^{?5}={2}^{5}=32\phantom{\rule{1em}{0ex}}$[?$\left(\frac{a}{b}{\right)}^{?n}=\left(\frac{b}{a}{\right)}^{n}$]

2.Simplify and express the result in power notation with a positive exponent. (i)$\left(?4{\right)}^{5}÷\left(?4{\right)}^{8}$
(ii)$\left(\frac{1}{{2}^{3}}{\right)}^{2}$

(iii)$\left(?3{\right)}^{4}×\left(\frac{5}{3}{\right)}^{4}$

(iv)$\left({3}^{?7}÷{3}^{?10}\right)×{3}^{?5}$

(v)${2}^{?3}×\left(?7{\right)}^{?3}$

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

(i)We have:$\left(?4{\right)}^{5}÷\left(?4{\right)}^{8}$
$=\left(?4{\right)}^{5?8}=\left(?4{\right)}^{?3}=\frac{1}{\left(?4{\right)}^{3}}\phantom{\rule{1em}{0ex}}$[?${a}^{m}÷{a}^{n}={a}^{m?n}$]

$=\left(?\frac{1}{4}{\right)}^{3}$

(ii)We have:$\left(\frac{1}{{2}^{3}}{\right)}^{2}$

$\left(\frac{1}{{2}^{3}}{\right)}^{2}=\frac{\left(1{\right)}^{2}}{\left({2}^{3}{\right)}^{2}}=\frac{1}{{2}^{6}}=\left(\frac{1}{2}{\right)}^{6}$

(iii)We have:$\left(?3{\right)}^{4}×\left(\frac{5}{3}{\right)}^{4}$

$=\left(?3{\right)}^{4}×\left(\frac{5}{3}{\right)}^{4}=\left(?3{\right)}^{4}×\frac{\left(5{\right)}^{4}}{\left(3{\right)}^{4}}$

$=\frac{\left(3{\right)}^{4}×\left(5{\right)}^{4}}{\left(3{\right)}^{4}}=\left(5{\right)}^{4}$

(iv)We have:$\left({3}^{?7}÷{3}^{?10}\right)×{3}^{?5}$

$=\left({3}^{?7}÷{3}^{?10}\right)×{3}^{?5}={3}^{?7?\left(?10\right)}×{3}^{?5}$

$={3}^{?7+10}×{3}^{?5}={3}^{3}×{3}^{?5}={3}^{\left(3?5\right)}$

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

(v)We have:${2}^{?3}×\left(?7{\right)}^{?3}$

$={2}^{?3}×\left(?7{\right)}^{?3}=\left[2×\left(?7\right){\right]}^{?3}=\left(?14{\right)}^{?3}$

$=?\left(14{\right)}^{?3}=?\frac{1}{{14}^{3}}=\left(?\frac{1}{14}{\right)}^{3}$

3.Find the value of
(i)${3}^{0}+{4}^{?1}×{2}^{2}$

(ii)${2}^{?1}×{4}^{?1}÷{2}^{?2}$

(iii)$\left(\frac{1}{2}{\right)}^{?2}+\left(\frac{1}{3}{\right)}^{?2}+\left(\frac{1}{4}{\right)}^{?2}$

(iv)$\left({3}^{?1}+{4}^{?1}+{5}^{?1}{\right)}^{0}$

(v)$\left[\left(\frac{?2}{3}{\right)}^{?2}{\right]}^{2}$

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

(i)We have:${3}^{0}+{4}^{?1}×{2}^{2}$

$={3}^{0}+{4}^{?1}×{2}^{2}=\left(1+\frac{1}{4}\right)×4$

$=\left(\frac{4+1}{4}\right)×4=\frac{5}{4}×4=5$

(ii)We have:${2}^{?1}×{4}^{?1}÷{2}^{?2}$

$={2}^{?1}×{4}^{?1}÷{2}^{?2}=\left(\frac{1}{2}×\frac{1}{4}\right)÷\frac{1}{{2}^{2}}$

$=\frac{1}{8}÷\frac{1}{4}=\frac{1}{8}×\frac{4}{1}$

$=\frac{1}{2}$

(iii)We have:$\left(\frac{1}{2}{\right)}^{?2}+\left(\frac{1}{3}{\right)}^{?2}+\left(\frac{1}{4}{\right)}^{?2}$

$=\left(\frac{1}{2}{\right)}^{?2}+\left(\frac{1}{3}{\right)}^{?2}+\left(\frac{1}{4}{\right)}^{?2}={2}^{2}+{3}^{2}+{4}^{2}$

$=4+9+16=29$

(iv)We have:$\left({3}^{?1}+{4}^{?1}+{5}^{?1}{\right)}^{0}$

$=\left({3}^{?1}+{4}^{?1}+{5}^{?1}{\right)}^{0}=1\phantom{\rule{1em}{0ex}}$[?${a}^{0}=1$]

(v)We have:$\left[\left(\frac{?2}{3}{\right)}^{?2}{\right]}^{2}$

$=\left[\left(\frac{?2}{3}{\right)}^{?2}{\right]}^{2}=\left[\left(?\frac{3}{2}{\right)}^{2}{\right]}^{2}$

$=\left(\frac{9}{4}{\right)}^{2}=\frac{81}{16}$

4.Evaluate
(i) $\frac{{8}^{?1}×{5}^{3}}{{2}^{?4}}$

(ii) $\left({5}^{?1}×{2}^{?1}\right)×{6}^{?1}$

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

(i)We have:$\frac{{8}^{?1}×{5}^{3}}{{2}^{?4}}$

$=\frac{{8}^{?1}×{5}^{3}}{{2}^{?4}}=\frac{1}{8}×{5}^{3}×{2}^{4}$

$=\frac{1}{8}×125×16=125×2=250$

(ii)We have:$\left({5}^{?1}×{2}^{?1}\right)×{6}^{?1}$

$=\left(\frac{1}{5}×\frac{1}{2}\right)×\frac{1}{6}$

$=\frac{1}{10}×\frac{1}{6}=\frac{1}{60}$

5.Find the value of m for which ${5}^{m}÷{5}^{?3}={5}^{5}$.

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

We have given: ${5}^{m}÷{5}^{?3}={5}^{5}$.

$?{5}^{m?\left(?3\right)}={5}^{5}\phantom{\rule{1em}{0ex}}$ [?${a}^{m}÷{a}^{n}={a}^{m?n}$]

$?{5}^{m+3}={5}^{5}$

Now, Comparing the powers of equal bases, we have m + 3 = 5

$?m=5–3=2$, i.e., m = 2

6.Evaluate:
(i)$\left[\left(\frac{1}{3}{\right)}^{?1}?\left(\frac{1}{4}{\right)}^{?1}{\right]}^{?1}$

(ii)$\left(\frac{5}{8}{\right)}^{?7}×\left(\frac{8}{5}{\right)}^{?4}$

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

(i)We have:$\left[\left(\frac{1}{3}{\right)}^{?1}?\left(\frac{1}{4}{\right)}^{?1}{\right]}^{?1}$

$=\left({3}^{1}?{4}^{1}{\right)}^{?1}=\left(?1{\right)}^{?1}=\frac{1}{\left(?1{\right)}^{1}}=\frac{1}{?1}=?1$

(ii)We have: $\left(\frac{5}{8}{\right)}^{?7}×\left(\frac{8}{5}{\right)}^{?4}$
$=\left(\frac{8}{5}{\right)}^{7}×\left(\frac{5}{8}{\right)}^{4}=\frac{{8}^{7}}{{5}^{7}}×\frac{{5}^{4}}{{8}^{4}}$

$=\frac{{8}^{7?4}}{{5}^{7?4}}=\frac{{8}^{3}}{{5}^{3}}=\frac{512}{125}$

7.Simplify :
(i)$\frac{25×{t}^{?4}}{{5}^{?3}×10×{t}^{?8}}\left(t?0\right)$

(ii)$\frac{{3}^{?5}×{10}^{?5}×125}{{5}^{?7}×{6}^{?5}}$

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

(i)We have:$\frac{25×{t}^{?4}}{{5}^{?3}×10×{t}^{?8}}\left(t?0\right)$

$=\frac{25×{5}^{3}}{10}×{t}^{\left(?4+8\right)}=\frac{5×{5}^{3}}{2}×{t}^{4}$

$=\frac{625}{2}{t}^{4}$
(ii)We have:$\frac{{3}^{?5}×{10}^{?5}×125}{{5}^{?7}×{6}^{?5}}$

$=\frac{{5}^{7}×{6}^{5}×125}{{3}^{5}×{10}^{5}}$

$=\frac{{5}^{7}×\left(3×2{\right)}^{5}×{5}^{3}}{{3}^{5}×\left(2×5{\right)}^{5}}=\frac{{5}^{7}×{3}^{5}×{2}^{5}×{5}^{3}}{{3}^{5}×{2}^{5}×{5}^{5}}={5}^{7+3?5}={5}^{5}$

NCERT solutions for class 8 Maths Chapter 12 Exponents And Powers Exercise 12.2

1.Express the following numbers in standard form:
(i) 0.0000000000085

(ii) 0.00000000000942

(iii) 6020000000000000

(iv) 0.00000000837

(v) 31860000000

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

(i)We have:0.0000000000085
$=\frac{85}{10000000000000}=\frac{8.5×10}{{10}^{13}}$

$=8.5×{10}^{\left(1?13\right)}=8.5×{10}^{?12}$

(ii)We have:0.00000000000942
$=\frac{942}{100000000000000}=\frac{9.42×{10}^{2}}{{10}^{14}}$

$=9.42×{10}^{\left(2?14\right)}=9.42×{10}^{?12}$

(iii)We have:6020000000000000=$6.02×1000000000000000$

$=6.02×{10}^{15}$

(iv)We have:0.00000000837

$=\frac{837}{100000000000}=\frac{8.37×100}{{10}^{11}}$

$=8.37×{10}^{\left(2?11\right)}=8.37×{10}^{?9}$

(v)We have: 31860000000

$=3.186×10000000000$

$=3.186×{10}^{1}0$

Hence, the required standard from = $3.186×{10}^{1}0$

2.Express the following numbers in usual form.
(i) $3.02×{10}^{?6}$

(ii) $4.5×{10}^{4}$

(iii) $3×{10}^{?8}$

(iv) $1.0001×{10}^{9}$

(v) $5.8×{10}^{12}$

(vi) $3.61492×{10}^{6}$

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

(i)We have:$3.02×{10}^{?6}$

$=\frac{302}{100}×\frac{1}{{10}^{6}}=\frac{302}{100000000}$

$=302×{10}^{?8}=0.00000302$

Thus, we have $3.02×{10}^{?6}=0.00000302$

(ii)We have:$4.5×{10}^{4}$

$=\frac{45}{10}×{10}^{4}=45×{10}^{3}$

$=45000$

Thus, we have $4.5×{10}^{4}=45000$

(iii)We have:$3×{10}^{?8}$

$=\frac{3}{{10}^{8}}=\frac{3}{100000000}$

$=0.00000003$

Thus, we have $3×{10}^{?8}=0.00000003$

(iv)We have: $1.0001×{10}^{9}$

$=\frac{10001}{10000}×{10}^{9}=\frac{10001}{{10}^{4}}×{10}^{9}$

$=10001×{10}^{5}=1000100000$

Thus, we have$1.0001×{10}^{9}=1000100000$

(v)We have: $5.8×{10}^{12}$

$=\frac{58}{10}×{10}^{12}=58×{10}^{11}$

$=5800000000000$

Thus, we have $5.8×{10}^{12}=5800000000000$

(vi)We have: $3.61492×{10}^{6}$

$=\frac{361492}{100000}×{10}^{6}=\frac{361492}{{10}^{5}}×{10}^{6}$

$=361492×10=3614920$

Thus, we have $3.61492×{10}^{6}=3614920$

3.Express the number appearing in the following statements in standard form.
(i) 1 micron is equal to $\frac{1}{1000000}$ m.

(ii) Charge of an electron is 0.000,000,000,000,000,000,16 coulomb

(iii) Size of a bacteria is 0.0000005 m

(iv) Size of a plant cell is 0.00001275 m

(v) Thickness of a thick paper is 0.07 mm.

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

(i)We have 1 micron=$\frac{1}{1000000}$

$=\frac{1}{{10}^{6}}m={10}^{?6}$

(ii)We know charge of an electron=0.000,000,000,000,000,000,16

$\frac{16}{1,000,000,000,000,000,000,00}=\frac{1.6×10}{1,000,000,000,000,000,000,00}$

$=\frac{1.6×10}{{10}^{20}}=1.6×{10}^{1?20}=1.6×{10}^{?19}$

(iii)We have size of bacteria=0.0000005m

$=\frac{5}{10000000}m=\frac{0.5×10}{{10}^{7}}m$

$=0.5×{10}^{1?7}m=0.5×{10}^{?6}m$

$=5×{10}^{?7}m$

(iv)We have size of plant cell=0.00001275m

$=\frac{1275}{100000000}m=\frac{1.275×{10}^{3}}{{10}^{8}}m$

$=1.275×{10}^{3?8}m=1.275×{10}^{?5}m$

$=1.275×{10}^{?5}m$

(v)We have thickness of paper=0.07mm

$=\frac{7}{100}mm=\frac{0.7×10}{{10}^{2}}mm$

$=0.7×{10}^{1?2}mm=0.7×{10}^{?1}mm$

$=7×{10}^{?2}mm$

4.In a stack there are 5 books each of thickness 20 mm and 5 paper sheets each of thickness 0.016 mm. What is the total thickness of the stack?

NCERT Solutions for Class 8 Maths Chapter 12 Exponents And Powers

We have the thickness of books = $5×20=100mm$

Also, the thickness of 5 paper sheets = $5×0.016mm=0.080mm$.

Total thickness of the stack = $100mm+0.080mm=100.080mm=1.0008×{10}^{2}mm$

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