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
$\begin{array}{l|l}5 & 8400 \\\hline 5 & 1680 \\\hline 3 & 336 \\\hline 2 &112 \\\hline 2 & 56\\\hline2&28\\\hline 2&14 \\\hline &7\end{array}$
$=2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5 \times 7$
$=2^{4} \times 3^{1} \times 5^{2} \times 7$
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