EXERCISE:4.1
Question 1
i) $3^{7}$
Base $=3 \quad ;$ Exponent $=7$
ii) $(-7)^{5}$
Base $=-7 ;$ Exponent $=5$
iii) $\left(\frac{2}{5}\right)^{11}$
Base $=\frac{2}{5}$; exponent $=11$
iv) Base $=6$; Exponent $=8$
Exponent form = Base exponent $=6^{8}$
Question 2
i) $2^{6}=2 \times 2 \times 2 \times 2 \times 2 \times 2=64$
ii) $5^{5}=5 \times 5 \times 5 \times 5 \times 5=3125 .$
iii) $(-6)^{4}=-6 x-6 x-6 x-6=1296$
iv) $\left(\frac{2}{3}\right)^{4}=\frac{2^{4}}{3^{4}}=\frac{2 \times 2 \times 2 \times 2}{3 \times 3 \times 3 \times 3}=\frac{16}{81}$
v) $\left(-\frac{2}{3}\right)^{5}=\frac{(-2)^{5}}{3^{5}}=\frac{-2 \times-2 \times-2 \times -2 \times -2}{3 \times 3 \times 3 \times 3 \times 3}=-\frac{32}{729}$
vi) $\quad(-2)^{9}=-2 \times-2 \times-2 \times-2 \times-2 \times-2 \times-2 \times-2 \times-2$
=512
Question 3
i) $6 \times 6 \times 6 \times 6 \times 6=6^{5}$
ii) $t \times t \times t=t^{3}$
(iii) 2 $\times$ 2 $\times$ a $\times$ a $\times$ a $\times$ a $=2^{2} \times a^{4}$
iv) $\operatorname{a\times a\times a\times }c \times c \times c \times c=a^{3} \times c^{4} \times d^{1}$
Question 4
i) $7 \times 10^{3}=7 \times 1000=7000$
ii) $2^{5} \times 9=2 \times 2 \times 2 \times 2 \times 2 \times 9=288$
iii) $3^{3} \times 10^{4}=3 \times 3 \times 3 \times 10 \times 10 \times 10 \times 10=270,000$
Question 5
i)
$\begin{aligned} &(-3) \times(-2)^{3} \\ &=-3 x-2 x-2 x-2 \\ &=24 \end{aligned}$
ii) $(-3)^{2} \times(-5)^{2}$
$=-3 \times -3 \times -5 \times -5$
$=225$
iii)
$\begin{aligned}(&-2)^{3} \times(-10)^{4} \\ &=-2 \times -2 \times-2 \times -10 \times -10 \times -10 \times -10 \\ &=-80,000 \end{aligned}$
iv) $(-1)^{9}=-1 \times -1 \times -1 \times -1 \times -1 \times -1 \times -1 \times -1 \times -1=-1$
v) $25^{2} \times(-1)^{31}=25 \times 25 \times -1=-625$
Question 6
i) $4^{3}=4 \times 4 \times 4=64 ; 3^{4}=3 \times 3 \times 3 \times 3=81$
$\therefore 3^{4}$ is greater
(ii) $7^{3}$ or $3^{\text {7 }}$
$7^{3}=7 \times 7 \times 7=343 ; 3^{7}=3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3=2187$
$\therefore 3^{7}$ is greater
iii) $4^{5}=4 \times 4 \times 4 \times 4 \times 4=1024 ; 5^{4}=5 \times 5 \times 5 \times 5=625$
$\therefore 4^{5}$ is greater
iv) $2^{10}=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2=1024 $
;$10^{2}=10 \times 10-100$
$\therefore 2^{10}$ is greater.
Question 7
i) 8
$\begin{array}{r|l}2&8\\ \hline 2& 4 \\ \hline&2\end{array}$
$=2 \times 2 \times 2$
$=2^{3}$
ii) 128
$\begin{array}{r|l}2&128\\ \hline 2& 64 \\ \hline 2&32\\ \hline 2& 16\\ \hline 2& 4 \\\hline&2\end{array}$
$-2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
$=2^{7}$
iii) 1024
$\begin{array}{r|l}2&1024\\ \hline 2& 512 \\ \hline 2&256\\ \hline 2& 128\\ \hline 2& 64 \\ \hline 2&32\\ \hline2&16 \\ \hline 2&8\\ \hline 2&4 \\ \hline&2\end{array}$
$=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
$=2^{10}$
Question 8
Let
$\begin{aligned}(-2)^{x} &=16 \\(-2)^{x} &=(-2)^{4} \end{aligned}$
Base is equal So, exponent should be same
i.e x =4
∴ Up to 4 should be raised
Question 9
i) $9=-3 \times -3=(-3)^{2}$
ii) $-27=-3 \times-3 \times-3=(-3)^{3}$
iii) $81=-3 \times-3 \times-3 \times-3=(-3)^{4}$
Question 10
i)
$\begin{aligned} 7^{x} &=343 \\ & 3^{x}=3^{6} \end{aligned}$
Base equal , exponent is same
i.e x= 6
ii) $3^{x}=729$
$\begin{array}{r|l}3&729\\ \hline 3& 243 \\ \hline 3&81\\ \hline 3& 27\\ \hline 3& 9 \\\hline&3\end{array}$
$3^{x}=3^{6}$
Base equal , exponent is same
i.e x = 6
iii) $(-8)^{x}=-512$
$(-8)^{x}=-8 \times -8 \times-8$
$\begin{array}{r|l}8&512\\ \hline 8& 64 \\ \hline&8\end{array}$
$(-8)^{x}=(-8)^{3}$
Base is equal , exponent should be same
x = 3
iv) $(-4)^{x}=-1024$
$(-4)^{x}=-4 \times -4 \times -4 \times -4 \times -4$
$\begin{array}{r|l}4&1024\\ \hline 4& 256 \\ \hline 4&64\\ \hline 4& 16\\ \hline&4\end{array}$
$(-4)^{x}=(-4)^{5}$
Base is equal , exponent should be same
x = 5
v) $\left(\frac{2}{5}\right)^{x}=\frac{32}{3125}$
$\left(\frac{2}{5}\right)^{x}=\frac{2 \times 2 \times 2 \times 2 \times 2}{5 \times 5 \times 5 \times 5 \times 5}$
$\left(\frac{2}{5}\right)^{x}=\frac{2^{5}}{5^{5}}$
$\left(\frac{2}{5}\right)^{x}=\left(\frac{2}{5}\right)^{5}$
Base equal , exponent should be same
i.e x = 5
vi) $\left(\frac{-3}{4}\right)^{x}=\frac{-243}{1024}$
$\left(-\frac{3}{4}\right)^{x}=\frac{-3 \times-3 \times-3 \times-3 \times-3}{4 \times 4 \times 4 \times 4 \times 4}$
$\left(-\frac{3}{4}\right)^{x}=\frac{(-3)^{5}}{(4)^{5}}$
$\left(\frac{-3}{4}\right)^{x}=\left(\frac{-3}{4}\right)^{5}$
Base is equal, exponent should be same
ஃ x = 5
Question 11
i) 72
$\begin{array}{r|l}2&72\\ \hline 2& 36 \\ \hline 2&18\\ \hline 3& 9\\ \hline&3\end{array}$
$=2 \times 2 \times 2 \times 3 \times 3$
$=2^{3} \times 3^{2}$
ii) 360
$\begin{array}{r|l}2&360\\ \hline 2& 180 \\ \hline 2&90\\ \hline 2& 45\\ \hline3&45\\ \hline 5\end{array}$
$=2 \times 2 \times 2 \times 3 \times 3 \times 5$
$=2^{3} \times 3^{2} \times 5^{1}$
iii) 405
$\begin{array}{r|l}5&405\\ \hline 3& 81 \\ \hline 3&27\\ \hline 3& 9\\ \hline 3\end{array}$
$3\times 3\times 3\times 3\times 5$
$=3^{4} \times 5^{1}$
iv) 540
$\begin{array}{r|l}3&540\\ \hline 3& 180 \\ \hline 3&60\\ \hline 3& 20\\ \hline2&4\\ \hline 2\end{array}$
$=2 \times 2 \times 3 \times 3 \times 3 \times 5$
$=2^{2} \times 3^{3} \times 5^{1}$
v) 2280 .
$\begin{array}{r|l}2&2280\\ \hline 2& 1140 \\ \hline 2&570\\ \hline 5& 285\\ \hline3&57\\ \hline 19\end{array}$
$=2 \times 2 \times 2 \times 3 \times 5 \times 19$
$=23 \times 3 \times 5^{1} \times 19^{1}$
vi) 3600
$\begin{array}{l|l}3 & 3600 \\\hline 3 & 1200 \\\hline 2 & 400 \\\hline 2 & 200 \\\hline 2 & 100 \\\hline 5 & 50 \\\hline&5\end{array}$
$=2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5$
$=2^{4} \times 3^{2} \times 5^{2}$
vii) 4725
$\begin{array}{l|l}5 & 4725 \\\hline 5 & 945 \\\hline 3 & 189 \\\hline 3 & 63 \\\hline 3 & 21\\\hline&7\end{array}$
$=3 \times 3 \times 3 \times 5 \times 5 \times 7$
$=3^{3} \times 5^{2} \times 7$
viii) 8400
$\begin{array}{l|l}5 & 8400 \\\hline 5 & 1680 \\\hline 3 & 336 \\\hline 2 &112 \\\hline 2 & 56\\\hline2&28\\\hline 2&14 \\\hline &7\end{array}$
$=2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5 \times 7$
$=2^{4} \times 3^{1} \times 5^{2} \times 7$
EXERCISE:4.2
Question 1
As we know that $a^{m} \times a^{n}=a^{m+n}$
i) $2^{7} \times 2^{4}$
$=2^{7+4}=2^{11}$
ii) $p^{5} \times p^{3}=p^{5+3}=p^{8}$
iii) iii. $(-7)^{5} \times(-7)^{n}=(-7)^{5+11}=(-7)^{16}$
iv) $\left(\frac{3}{5}\right)^{6} \div\left(\frac{3}{5}\right)^{2}$
As we know $a^{m} \div a^{n}=a^{m-n}$
i.e $=\left(\frac{3}{5}\right)^{6-2}=\left(\frac{3}{5}\right)^{4}$
v)
$\begin{aligned}(-6)^{7} \div(-6)^{3} \\=&(-6)^{7-3} \\=(-6)^{4} \end{aligned}$
Question 2
i) $5^{3} \times 5^{7} \times 5^{12}$
$=5^{3+7+12}$
$=5^{22}$
ii) $a^{5} \times a^{3} \times a^{7}$
$=a^{5+3+7}$
$=a^{15}$
iii)$\left(7^{12} \times 7^{3}\right) \div 7^{4}$
$=7^{12+3} \div 7^{4}$
$=7^{15} \div 7^{4}$
$=7^{15-4}$
$=7^{11}$
Question 3
i) $\left(2^{2}\right)^{100}$
As we $\left(a^{m}\right)^{n}=a^{m n}$
$=2^{2 \times 100}=2^{200}$
ii) $\left((-7)^{6}\right)^{5}$
$=(-7)^{6 \times 5}=(-7)^{30}$
iii) $\quad\left(3^{2}\right)^{5} \times\left(3^{4}\right)^{7}$
$=3^{2 \times 5} \times 3^{4 \times 7}$
$=3^{10} \times 3^{28}$
$=3^{10+28}$
$=3^{38}$
Question 4
i) $\frac{a^{3} \times a^{5}}{\left(a^{3}\right)^{2}}$
$=\frac{a^{3+5}}{a^{3 x^{2}}}$
$=\frac{a^{8}}{a^{6}}$
$=a^{8-6}=a^{2}$
ii) $\left(2^{3}\right)^{4} \div 2^{5}$
$=2^{3 \times 4} \div 2^{5}$
$=2^{12} \div 2^{5}$
$=2^{12-5}$
$=2^{7}$
iii. $\left[\left(6^{2}\right)^{3} \div 6^{3}\right] \times 6^{5}$
$=\left[6^{2 \times 3}+6^{3}\right] \times 6^{5}$
$=\left[6^{6} \div 6^{3}\right] \times 6^{5}$
$=6^{6-3} \times 6^{5}$
$=6^{3} \times 6^{5}$
$=6^{3+5}$
$=6^{8}$
Question 5
i) $5^{4} \times 8^{4}$
As we know that $a^{m} \times b^{m}=(a \times b)^{m}=(a b)^{m}$
$=(5 \times 8)^{4}$
$=40^{4}$
ii)
$\begin{aligned} &(-3)^{6} \times(-5)^{6} \\ &=(-3 x-5)^{6} \\ &=15^{6} \end{aligned}$
iii) $\left(\frac{3}{10}\right)^{5} \times\left(\frac{2}{15}\right)^{5}$
$=\left[\frac{3}{10} \times \frac{2}{15}\right]^{5}$
$=\left[\frac{1}{25}\right]^{5}$
Question 6
$\frac{2^{4} \times 2 \times 3^{3} \times 7}{2^{3} \times 7^{4}}$
$=\frac{2^{4+1} \times 3^{3+6}}{2^{3} \times 3^{4}}$
$=\frac{2^{5}}{2^{3}} \times \frac{7^{9}}{7^{4}}$
$=2^{5-3} \times 9^{9-4}$
$=2^{2} \times 7^{5}$
ii) $\left(3^{2}\right)^{3} \times(-2)^{5}$
$\frac{-3^{3 \times 3} \times(-2)^{5}}{(-2)^{3}}$
$=3^{6} \times \frac{(-2)^{5}}{(-2)^{3}}$
$=3^{6} \times(-2)^{5-3}$
$=3^{6} \times(-2)^{2}$
iii) $\frac{2^{8} \times a^{5}}{4^{3} \times a^{3}}$
As $4=2^{2}$
$=\frac{2^{8} \times a^{5}}{\left(2^{2}\right)^{3} \times a^{3}}$
$=\frac{2^{8}}{2^{6}} \times \frac{a^{5}}{a^{3}}$
$=2^{8-6} \times a^{5-3}$
$=2^{2} \times a^{2}=(2 \times a)^{2}=4 a^{2}$
iv)$\frac{3 \times 7^{2} \times 11^{8}}{21 \times 11^{3}}$
As $21=7 \times 3$
$=\frac{3 \times 7^{2} \times 11^{8}}{3 \times 7 \times 11}$
$=\frac{7^{2}}{7^{1}} \times \frac{11^{8}}{11^{1}}$
$=7^{2-1} \times 11^{8-1}$
$=7^{1} \times 11^{7}$
v) $\left(2^{0}+3^{0}\right) 4$
As we rnow $a^{\circ}=1$
i.e $2^{\circ}=1 ; 3^{\circ}=1: 4^{\circ}=1$
$=(1+1) \times 1=2 \times 1=2$
vi)$3^{\circ} \times 4^{\circ} \times 5^{\circ}$
As we know $a^{\circ}=1$
$3^{0}=1 \quad ; 4^{\circ}=1 ; 5^{\circ}=1$
$=1\times1\times 1$
$=1$
Question 7
i) $\frac{25}{64}$
$\begin{array}{r|l}2&64\\ \hline 2& 32 \\ \hline 2&16\\ \hline 2& 8\\ \hline 2&4\\ \hline &2 \end{array}$
$=2 \times 2 \times 2 \times 2 \times 2 \times 2$
$=2^{6}$
$\begin{array}{r|l}5&25\\ \hline &5\end{array}$
$=5 \times 5$
$=5^{2}$
$=\frac{5^{2}}{2^{6}}$
ii. $\frac{-125}{216}$
$\begin{array}{r|l}5&125\\ \hline 5&25\\ \hline &5\end{array}$
$=5 \times 5 \times 5$
$=5^{3}$
$\begin{array}{r|l}2&216\\ \hline 2&72 \\ \hline 2&36\\ \hline 2& 18\\ \hline 2&9\\ \hline &3 \end{array}$
$=2 \times 2 \times 2 \times 3 \times 3 \times 3$
$=2^{3} \times 3^{3}=(3 \times 2)^{3}$
$=6^{3}$
$=\frac{5^{3}}{6^{3}}$
$=\left(\frac{-5}{6}\right)^{3}$
iii $\frac{-343}{729}$
$\begin{array}{r|l}7&343\\ \hline 7&49\\ \hline &7\end{array}$
$=7 \times 7 \times 7$
$=7^{3}$
$\begin{array}{r|l}9&729\\ \hline 9&81\\ \hline &9\end{array}$
$=9 \times 9 \times 9$
$=9^{3} .$
$=\frac{-7^{3}}{9^{3}}$
$=\left(\frac{-7}{9}\right)^{3}$
Question 8
i)
$\begin{aligned} & \frac{\left(2^{5}\right)^{2} \times 7^{3}}{8^{3} \times 3} \\=& \frac{2^{5 \times 2} \times 3^{3}}{\left(2^{3}\right)^{3} \times 7} \\=& \frac{2^{10} \times 7^{3}}{2^{9} \times 7^{1}} \end{aligned}$
$=2^{10-9} \times 3^{3-1}$
$2^{1} \times 3^{2}=2 \times 49=98$
ii) $\frac{25 \times 5^{2} \times t^{8}}{10^{3} \times t^{4}}$
$10=5 \times 2 ; 25=5 \times 5=5^{2}$
$\frac{5^{2} \times 5^{2} \times t^{8}}{(5 \times 2)^{3} \times t^{4}}$
$=\frac{5^{2+2} \times t^{8}}{5^{3} \times 2^{3} \times t^{4}}$
$=\frac{5^{4}}{5^{3}} \times \frac{t^{8}}{t^{4}} \times \frac{1}{2^{3}}$
$=\frac{5^{4-3} \times t^{8-4}}{2^{3}}$
$\frac{5 \times t^{4}}{2^{3}} \cdot=\frac{5 \times t^{4}}{8}$
iii) $\frac{3^{5} \times 10^{5} \times 25}{5^{7} \times 6^{5}}$
As $10=5 \times 2 ; 25=5 \times 5 ; 6=3 \times 2$
$=\frac{3^{5} \times(5 \times 2)^{5} \times 5 \times 5}{5^{7} \times 6^{5}}$
$=\frac{3 \times 5^{5} \times 2^{5} \times 5^{2}}{5^{7} \times(3 \times 2)^{5}}$
$=\frac{3^{5} \times 2^{5} \times 5^{5+2}}{5^{7} \times 3^{5} \times 2^{5}}$
$=\frac{3^{5}}{3^{5}} \times \frac{2^{5}}{2^{5}} \times \frac{5^{7}}{5^{7}}$
$=3^{5-5} \times 2^{5-5} \times 5^{7-7}$
$=3^{0} \times 2^{\circ} \times 5^{\circ}$
$=1 \times 1 \times 1=1$
iv) $\left(\frac{-3}{5}\right)^{-3}$
As we know $a^{-n}=\frac{1}{a^{n}}$
$=\frac{1}{\left(\frac{-3}{5}\right)^{3}}$
$=\left(\frac{-5}{3}\right)^{3}$
$=\frac{(45)^{3}}{(3)^{3}}$
$=\frac{-5 \times -5 \times-5}{3 \times 3 \times 3}$
$=\frac{-125}{27}$
Question 9
i)
$\begin{aligned} &\left[\frac{-1}{2}\right]^{5} \times 2^{6} \times\left(\frac{3}{4}\right)^{3} \\=& \frac{(-1)^{5} \times 2^{6} \times 3^{3}}{(-2)^{5} \times 4^{3}} \end{aligned}$
$2^{6}=2 \times 2 \times 2 \times 2 \times 2 \times 2$
$=64$
$3^{3}=3 \times 3 \times 3=27$
$2^{5}=2 \times 2 \times 2 \times 2 \times 2=32$
$4^{3}=4 \times 4 \times 4=64$
$=\frac{1 \times 64 \times 27}{32 \times 64}$
$=\frac{27}{32}$
ii) $\left[\left(\frac{-3}{4}\right)^{3} \div\left(-\frac{5}{2}\right)^{3}\right] \times\left(-\frac{2}{3}\right)^{4}$
$=\left(\frac{3}{42} \times \frac{2}{5}\right)^{3} \times\left(-\frac{2}{3}\right)^{4}$
$=\frac{3^{3}}{10^{3}} \times \frac{-2 \times -2 \times-2 \times -2}{3 \times 3 \times 3 \times 3}$
$\frac{27 \times 16}{1000 \times 81}$
$=\frac{2}{375}$
Question 10
i) $\left(\frac{3}{2}\right)^{-1} \div\left(-\frac{2}{5}\right)^{-1}$
$\left(\frac{3}{2}\right)^{-1}=\frac{1}{\left(\frac{3}{2}\right)}=\frac{2}{3}$
$\left(\frac{-2}{5}\right)^{-1}=\frac{1}{\left(\frac{-2}{5}\right)}=\frac{5}{-2}$
$=\frac{2}{3} \times \frac{5}{-2}$
$=\frac{-5}{3}$
ii. $\left[\left\{\left(\frac{-1}{4}\right)^{2}\right\}^{-1}\right]^{-2}$
$=\left(-\frac{1}{4}\right)^{2 \times-1 \times-2}$
$\left(\frac{-1}{4}\right)^{4}=\frac{(-1)^{4}}{4^{4}}$
$=\frac{-1 \times-1 \times -1 \times-1}{4 \times 4 \times 4\times 4}$
$=\frac{1}{256}$
Question 11
$\left(\frac{1}{3}\right)^{-2}+\left(\frac{1}{4}\right)^{-2}+\left(\frac{1}{5}\right)^{-2}-\left(\frac{1}{6}\right)^{-2}$
As we know $a^{-n}=\frac{1}{a^{n}}$
$\frac{1}{\left(\frac{1}{3}\right)^{2}}+\frac{1}{\left(\frac{1}{4}\right)^{2}}+\frac{1}{\left(\frac{1}{5}\right)^{2}}-\frac{1}{\left(\frac{1}{6}\right)^{2}}$
$=3^{2}+4^{2}+5^{2}-6^{2}$
$=9+16+25-36$
$=50-36$
$=14 .$
Question 12
i) $108 \times 192$
$\begin{array}{r|l}3&108\\ \hline 3&36 \\ \hline 3&12\\ \hline 2& 4\\ \hline &2 \end{array}$
$\begin{aligned} 108 &=3 \times 3 \times 3 \times 2 \times 2 \\ &=2^{2} \times 3^{3} \end{aligned}$
$\begin{array}{r|l}3&192\\ \hline 2&64\\ \hline 2&32\\ \hline 2& 16\\ \hline 2&8\\ \hline 2&4\\ \hline&2 \end{array}$
$\begin{aligned} 192=& 2 \times 2 \times 2 \times 2 \times 2 \times 3 \\ &=2^{5} \times 3^{1} \end{aligned}$
$\begin{aligned} 108 \times 192=& 2^{2} \times 3^{3} \times 2 \times 3^{5} \times 1 \\ &=2^{2+5} \times 3^{3+1} \\ &=2^{7} \times 3^{4} \end{aligned}$
ii)$729 \times 64$
$\begin{array}{r|l}3&729\\ \hline 3&243\\ \hline 3&81\\ \hline 3& 27\\ \hline 3&9\\ \hline&3 \end{array}$
$729=3 \times 3 \times 3 \times 3 \times 3 \times 3=3^{6}$
$\begin{array}{r|l}2&64\\ \hline 2&32\\ \hline 2&16\\ \hline 2& 8\\ \hline 2&4\\ \hline 2&2\\ \hline&1 \end{array}$
$64=2 \times 2 \times 2 \times 2 \times 2 \times 2=2^{6}$
$729 \times 64=2^{6} \times 3^{6}$
iii) $384 \times 147$
$\begin{array}{r|l}3&384\\ \hline 2&128\\ \hline 2&64\\ \hline 2&32\\ \hline 2&16\\ \hline 2&8\\ \hline2&4\\ \hline &2 \end{array}$
$\begin{aligned} 384 &=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ &=2^{7} \end{aligned}$
$\begin{array}{r|l}3&147\\ \hline 7&49\\ \hline7 &7\\ \hline &1\end{array}$
$\begin{aligned} 147 &=3 \times 7 \times 7 \\ &=3 \times 7^{2} \end{aligned}$
$384 \times 147=2^{7} \times 3 \times 7^{2}$
Question 13
i) $3^{3} \times 2^{2}+2^{2} \times 5^{\circ}$
$\begin{array}{r|l}2&112\\ \hline 2&56\\ \hline 2&28\\ \hline 2&14\\ \hline 2&7\\ \hline &7 \end{array}$
$=3 \times 3 \times 3 \times 2 \times 2+2 \times 2 \times 1$
$=108+4$
$=112$
$=2 \times 2 \times 2 \times 2 \times 7$
$=2^{4} \times 7^{1}$
ii)
$\begin{aligned} & 9^{2}+11^{2}-2^{2} \times 3 \times 17^{\circ} \\ & 2^{2} \times 3 \times 17^{\circ}=4 \times 3 \times 1=12 \end{aligned}$
$9^{2}=9 \times 9=81$
$11^{2}$ = $11\times 11$ = 121
$=81+121-12$
$=202-12$
$=190 .$
Question 14
i) Let the number multiplied be x
$\therefore \quad x \times 3^{4}=3^{7}$
$x=\frac{3^{3}}{3^{4}}$
$x=3^{7-4}$
$x=3^{3}=3 \times 3 \times 3$
$x=27$
ஃ 27 should be multiplied in order to get $3^{7}$
ii) Let the number multiplied be x
$(-6)^{-1} \times x=10^{-1}$
$\frac{1}{-6} \times x=\frac{1}{10}$
$x=\frac{-6}{10}$
$x=\frac{-3}{5}$
So, $\frac{-3}{5}$ Should be multiplied in order to get $10^{-1} .$
Question 15
$\left(\frac{12}{13}\right)^{4} \times\left(\frac{13}{12}\right)^{-8}=\left(\frac{12}{13}\right)^{2 x}$
$\left(\frac{12}{13}\right)^{4} \times \frac{1}{\left(\frac{13}{12}\right)^{8}}=\left(\frac{12}{13}\right)^{2 x}$
$\left(\frac{12}{13}\right)^{4} \times\left(\frac{12}{13}\right)^{8}=\left(\frac{12}{13}\right)^{2 x}$
$\left(\frac{12}{13}\right)^{4+8}=\left(\frac{12}{13}\right)^{2 x}$
$\left(\frac{12}{13}\right)^{12}=\left(\frac{12}{13}\right)^{2 x}$
Base equal , exponent should be same
2 x=12
∴ x=6
EXERCISE:4.3
Question 1
i. $530.7=5.367 \times 10^{2}$
ii) $3908 \cdot 78=3.90878 \times 10^{3}$
iii) $39087.8=3.90878 \times 10^{4}$
iv) 2.35 it is a standard form
v) $3,43,000=3.43000 \times 10^{5}$
vi) $70,00,000=7.000000 \times 10^{6}$
vii). $3,18,65,00,000=3.186500000 \times 10^{9}$
Viii) $893,000,000=8.93000000 \times 10^{8}$
ix) $70,040,000,000=7.0040000000 \times 10^{10}$
Question 2
1) $4.7 \times 10^{3}=4700$
ii) $1.205 \times 10^{5}=120500$
iii) $1.234 \times 10^{6}=1234000$
iv) $4.87 \times 10^{7}=48700000$
V) $6.05 \times 10^{8}=605000000$
vi) $9.083 \times 10^{11}=908300000000$
Question 3
i) The distance between earth and the man is 384,00,0000
= $3.84 \times 10^{8} \mathrm{~m}$
ii) The diameter of sun$=1,400,000,000$
$=1.4 \times 10^{9} \mathrm{~m}$
iii) The universe is estimated to be about 12,000,000,000 year old
$=1.2 \times 10^{10}$ years old
iv) In a galaxy there are on an average 100,000,000,000 stars
$=1.0 \times 10^{11}$ stars
v) The distance of sun from the centre of milky way is 300,000,000,000,000,000
$=3 \times 10^{10} \mathrm{~km}$
vi) A light years is about 9,467,500,000,000km
$=9.4635 \times 10^{12} \mathrm{~km}$
Question 4
i) $4.3 \times 10^{14} ; 3.01 \times 10^{17}$
The given numbers are $4.3 \times 10^{14}$ and $3.01\times 10^{17}.$ . Note that both the numbers are in Standard form
Since the power of 10 in 4.3 $\times 10^{14}$ is less than power of 10 in 3.01$\times $ $10^{17} .$
$\therefore 4.3 \times 10^{14}<3.01 \times 10^{17}$
ii) The given numbers are $1.439 \times 10^{12}$ and $1.4335 \times 10^{12}$ . Note that both the numbers are in Standard form. Also note that both the numbers have same power of 10
So, we compare their significants
The significands in $1.439 \times 10^{12}$ is 1.439 and the significand in $1.4335 \times 10^{12}$ is $1.4335 .$
As, $1.439>1.4335$, So
$\therefore \quad 1.439 \times 10^{12}>1.4335 \times 10^{12}$
Question 5
i)279404
$\begin{aligned} 279404=& 2 \times 100000+7 \times 10000+9 \times 1000+4 \times 100+0 \times 10 \\ &+4 \times 1 \\=& 2 \times 10^{5}+7 \times 10^{4}+9 \times 10^{3}+4 \times 10^{2}+0 \times 10^{1}+4 \times 10^{\circ} \\=& 2 \times 10^{5}+7 \times 10^{4}+9 \times 10^{3}+4 \times 10^{2}+4 \times 10^{1} . \end{aligned}$
ii)3006194
$\begin{aligned} 3006194=& 3 \times 1000000+0 \times 100000+0 \times 10000+6 \times 1000+1 \times 100 \\ &+9 \times 10+4 \times 1 \\=& 3 \times 10^{6}+0 \times 10^{5}+0 \times 10^{4}+6 \times 10^{3}+1 \times 10^{2}+9 \times 10^{1}+4 \times 10^{\circ} \\=& 3 \times 10^{6}+6 \times 10^{3}+1 \times 10^{2}+9 \times 10^{\prime}+4 \times 10^{\circ} \end{aligned}$
iii)28061906
$\begin{aligned}=& 2 \times 10000000+8 \times 1000000+0 \times 100000+6 \times 10000+ \\ & 1 \times 1000+9 \times 100+0 \times 10+6 \times 1 \\=& 2 \times 10^{7}+8 \times 10^{6}+0 \times 10^{5}+6 \times 10^{4}+1 \times 10^{3}+9 \times 10^{2} \\ &+0 \times 10^{1}+6 \times 10^{\circ} \\=& 2 \times 10^{\circ}+8 \times 10^{6}+6 \times 10^{4}+1 \times 10^{3}+9 \times 10^{2}+6 \times 10^{\circ} . \end{aligned}$
Question 6
i) $3 \times 10^{4}+7 \times 10^{2}+5 \times 10^{\circ}$
$=3 \times 10000+7 \times 100+5 \times 1$
$=30000+700+5$
$=30705$
ii)
$\begin{aligned} & 4 \times 10^{5}+5 \times 10^{3}+3 \times 10^{2}+2 \times 10^{\circ} \\=& 4 \times 100000+5 \times 1000+3 \times 100+2 \times 1 \\=& 400000+5000+300+2 \\=& 405302 \end{aligned}$
iii) $8 \times 10^{7}+3 \times 10^{4}+7 \times 10^{3}+5 \times 10^{2}+8 \times 10^{1}$
$=8 \times 10000000+3 \times 10000+7 \times 1000+5 \times 100+80$
$=80000000+30000+7000+500+80$
=80037580
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