Exercise 1.6 - Chapter 1 Numbers 8th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Question $1 .$
Fill in the blanks.
(i) $(-1)^{\text {even integer }}$ is
Answer:
1
(ii) For $a \neq 0, a^{0}$ is
Answer:
1
(iii) $4^{-3} \times 5^{-3}=$
Answer:
$20^{-3}$
(iv) $(-2)^{-7}$ is $=$
Answer:
$\frac{-1}{128}$
(v) $\left(-\frac{1}{3}\right)^{-5}=$
Answer:
$-243$

Question $2 .$
Say True or False:
(i) If $8^{x}=\frac{1}{64}$, the value of $x$ is $-2$.
Answer:
True

(ii) The simplified form of $(256)^{\frac{-1}{4}} \times 4^{2}$ is $\frac{1}{4}$.
Answer:
True
(iii) Using the power rule, $\left(3^{7}\right)^{-2}=3^{5}$
Answer:
True
(iv) The standard form of $2 \times 10^{-4}$ is $0.0002$.
Answer:
False
(v) The scientific form of $123.456$ is $1.23456 \times 10^{-2}$.
Answer:
True
 

Question $3 .$
Evaluate
(i) $\left(\frac{1}{2}\right)^{3}$
(ii) $\left(\frac{1}{2}\right)^{-5}$
(iii) $\left(\frac{-5}{6}\right)^{-3}$
(iv) $\left(2^{-5} \times 2^{7}\right) \div 2^{-2}$
(v) $\left(2^{-1} \times 3^{-1}\right) \div 6^{-2}$

Answer:
(i) $\left(\frac{1}{2}\right)^{3}$
$\left(\frac{1}{2}\right)^{3}=\frac{1^{3}}{2^{3}}=\frac{1}{2 \times 2 \times 2}=\frac{1}{8}$
(ii) $\left(\frac{1}{2}\right)^{-5}$
$\left(\frac{1}{2}\right)^{-5}=\frac{1^{-5}}{2^{-5}}=\frac{1}{2^{-5}}=2^{5}=2 \times 2 \times 2 \times 2 \times 2=32$

$\begin{aligned}
&\text { (iii) }\left(\frac{-5}{6}\right)^{-3} \\
&\left(\frac{-5}{6}\right)^{-3}=\frac{(-5)^{-3}}{6^{-3}}=\frac{6^{3}}{(-5)^{3}}=\left(\frac{(-5 \times-5 \times-5)}{(6 \times 6 \times 6)}\right)=-\frac{216}{125} \\
&\text { (iv) }\left(2^{-5} \times 2^{7}\right) \div 2^{-2} \\
&\left(2^{-5} \times 2^{7}\right) \div 2^{-2}=\left(2^{-5+7}\right) \div 2^{-2} \\
&=2^{2} \div 2^{-2} \\
&=2^{2+2} \\
&=2^{4} \\
&=16 \\
&\text { (v) }\left(2^{-1} \times 3^{-1}\right) \div 6^{-2} \\
&\left(2^{-1} \times 3^{-1}\right) \div 6^{-2}=(2 \times 3)^{-1} \div 6^{-2} \\
&=\left(6^{-1}\right) \div 6^{-2} \\
&=6^{(-1)-(-2)} \\
&=6^{1} \\
&=6
\end{aligned}$

Question $4 .$
Evaluate
(i) $\left(\frac{2}{5}\right)^{4} \times\left(\frac{5}{2}\right)^{-2}$
(ii) $\left(\frac{4}{5}\right)^{-2} \div\left(\frac{4}{5}\right)^{-3}$
(iii) $2^{7} \times\left(\frac{1}{2}\right)^{-3}$

Answer:

$\begin{aligned}
&\text { (i) }\left(\frac{2}{5}\right)^{4} \times\left(\frac{5}{2}\right)^{-2} \\
&\left(\frac{2}{5}\right)^{4} \times\left(\frac{2}{5}\right)^{2}=\left(\frac{2}{5}\right)^{4+2}=\left(\frac{2}{5}\right)^{6}
\end{aligned}$
$\text { (ii) } \begin{aligned}
&\left(\frac{4}{5}\right)^{-2} \div\left(\frac{4}{5}\right)^{-3} \\
=&\left(\frac{4}{5}\right)^{-2} \times\left(\frac{4}{5}\right)^{3}=\left(\frac{4}{5}\right)^{-2+3}=\left(\frac{4}{5}\right)^{-2+3}=\left(\frac{4}{5}\right)^{1}=\frac{4}{5}
\end{aligned}$
(iii) $2^{7} \times\left(\frac{1}{2}\right)^{-3}$
$\begin{aligned}
&=2^{7} \times 2^{3} \\
&=2^{7+3} \\
&=2^{10}
\end{aligned}$

Question $5 .$
Evaluate:
(i) $\left(5^{0}+6^{-1}\right) \times 3^{2}$
(ii) $\left(2^{-1}+3^{-1}\right) \div 6^{-1}$
(iii) $\left(3^{-1}+4^{-2}+5^{-3}\right)^{0}$
Answer:
(i) $\left(5^{0}+6^{-1}\right) \times 3^{2}$
$\begin{aligned}
\left(5^{0}+6^{-1}\right) \times 3^{3} &=\left(5^{0} \times 3^{3}\right)+\left(6^{-1} \times 3^{3}\right)=(1 \times 27)+\left(\frac{1}{2 \times 3} \times 3^{3}\right) \\
&=27+\left(\frac{1}{2} \times 3^{3-1}\right)=27+\left(\frac{1}{2} \times 3^{2}\right)
\end{aligned}$

$=27+\frac{9}{2}=\frac{54+9}{2}=\frac{63}{2}$
(ii) $\left(2^{-1}+3^{-1}\right) \div 6^{-1}$
Answer:
$\begin{aligned}
&\left(2^{-1}+3^{-1}\right) \div 6^{-1}=\left(\frac{1}{2}+\frac{1}{3}\right)+6^{-1} \\
&=\left(\frac{3+2}{6}\right)+6^{-1}=\left(\frac{5}{6}\right)+6^{-1}=\frac{5}{6} \times 6=5
\end{aligned}$

(iii) $\left(3^{-1}+4^{-2}+5^{-3}\right)^{0}$
Answer:
$\left(3^{-1}+4^{-2}+5^{-3}\right)^{0}=1$
$\left[\because a^{0}=1\right.$ where $\left.a \neq 0\right]$

Question $6 .$
Simplify
(i) $\left(3^{2}\right)^{3} \times\left(2 \times 3^{5}\right)^{-2} \times(18)^{2}$
(ii) $\frac{9^{2} \times 7^{3} \times 2^{5}}{8^{3}}$
(iii) $\frac{2^{8} \times 2187}{3^{5} \times 3^{2}}$
Answer:
(i) $\left(3^{2}\right)^{3} \times\left(2 \times 3^{5}\right)^{-2} \times(18)^{2}$
$\left(3^{2}\right)^{3} \times\left(2 \times 3^{5}\right)^{-2} \times(18)^{2}=3^{2 \times 3} \times \frac{1}{\left(2 \times 3^{5}\right)^{2}} \times 18^{2}$
$=3^{6} \times \frac{1}{2^{2} \times\left(3^{5}\right)^{2}} \times 18^{2}=3^{6} \times \frac{1}{2^{2} \times 3^{10}} \times\left(2 \times 3^{2}\right)^{2}$
$=3^{6} \times \frac{1}{2^{2} \times 3^{10}} \times 2^{2} \times 3^{2 \times 2}=3^{6} \times \frac{1}{2^{2} \times 3^{10}} \times 2^{2} \times 3^{4}$
$=\frac{3^{6+4} \times 2^{2}}{2^{2} \times 3^{10}}=\frac{3^{10} \times 2^{2}}{2^{2} \times 3^{10}}=1$

(ii) $\frac{9^{2} \times 7^{3} \times 2^{5}}{84^{3}}$
$\begin{aligned}
\frac{9^{2} \times 7^{3} \times 2^{5}}{84^{3}} &=\frac{\left(3^{2}\right)^{2} \times 7^{3} \times 2^{5}}{\left(2^{2} \times 3 \times 7\right)^{3}}=\frac{3^{2 \times 2} \times 7^{3} \times 2^{5}}{2^{2 \times 3} \times 3^{3} \times 7^{3}}=\frac{3^{4} \times 7^{3} \times 2^{5}}{2^{6} \times 3^{3} \times 7^{3}} \\
&=3^{4-3} \times 7^{3-3} \times 2^{5-6}=3^{1} \times 7^{0} \times 2^{-1} \\
&=3 \times 1 \times 2^{-1}=3 \times \frac{1}{2}=\frac{3}{2}
\end{aligned}$
(iii) $\frac{2^{5} \times 2187}{3^{5} \times 3^{2}}$

$\begin{aligned}
&\frac{2^{8} \times 2187}{3^{5} \times 3^{2}}=\frac{2^{8} \times 3^{7}}{3^{5} \times 2^{5}} \\
&=2^{8-5} \times 3^{7-5} \\
&=2^{3} \times 3^{2} \\
&=8 \times 9 \\
&=72
\end{aligned}$

Question 7.
Solve for $\mathrm{x}$ :
(i) $\frac{2^{2 z-1}}{2^{z+2}}=4$
(ii) $\frac{5^{5} \times 5^{-4} \times 5^{z}}{5^{12}}=5^{-5}$
Answer:
(i) $\frac{2^{2 z-1}}{2^{z+2}}=4$
$\begin{aligned}
&2^{2 x-1-(x+2)}=2^{2} \\
&2^{2 x-1-x-2)}=2^{2} \\
&2^{2 x-3}=2^{2}
\end{aligned}$
Equating the powers of the same base 2 .
$x-3=2$
$x-3+3=2+3$

$x-3+3=2+3$ $x=5$ (ii) $\frac{5^{5} \times 5^{-4} \times 5^{z}}{5^{12}}=5^{-5}$ $\frac{5^{5} \times 5^{-4} \times 5^{x}}{5^{12}}=5^{-5} \Rightarrow \frac{5^{5-4+x}}{5^{12}}=5^{-5}$ $\quad \Rightarrow \frac{5^{1+x}}{5^{12}}=5^{-5}$ $\Rightarrow 5^{1+x-12}=5^{-5}$ $\Rightarrow 5^{x-11}=5^{-5}$ Equating the powers of same base $5 .$ $x-11=-5$ $x-11+11=-5+11$ $x=6$
Equating the powers of same base 5 .

Question 8.
Expand using exponents:
(i) $6054.321$
(ii) 897.14
Answer:
(i) $6054.321$
6054.321 $=(6 \times 1000)+(0 \times 100)+(5 \times 10)+\left(4 \times 10^{0}\right)+\frac{3}{10}+\frac{2}{100}+\frac{1}{1000}$
$\begin{aligned}
&=\left(6 \times 10^{3}\right)+\left(5 \times 10^{1}\right)+\left(4 \times 10^{0}\right)+\frac{3}{10}+\frac{2}{100}+\frac{1}{1000} \\
&=\left(6 \times 10^{3}\right)+\left(5 \times 10^{1}\right)+\left(4 \times 10^{0}\right)+\left(3 \times 10^{-1}\right)+\left(2 \times 10^{-2}\right)+\left(1 \times 10^{-3}\right)
\end{aligned}$

(ii) $897.14$
$\begin{aligned}
&=(8 \times 100)+(9 \times 10)+\left(7 \times 10^{0}\right)+\frac{1}{10}+\frac{4}{100} \\
&=\left(8 \times 10^{2}\right)+\left(9 \times 10^{1}\right)+\left(7 \times 10^{0}\right)+\left(1 \times \frac{1}{10}\right)+\left(4 \times \frac{1}{100}\right) \\
&=\left(8 \times 10^{3}\right)+\left(9 \times 10^{3}\right)+\left(7 \times 10^{0}\right)+\left(1 \times 10^{-1}\right)+\left(4 \times 10^{-2}\right)
\end{aligned}$

Question $9 .$
Find the number is standard form:
(i) $8 \times 10^{4}+7 \times 10^{3}+6 \times 10^{2}+5 \times 10^{1}+2 \times 1+4 \times 10^{-2}+7 \times 10^{-4}$
(ii) $5 \times 10^{3}+5 \times 10^{1}+5 \times 10^{-1}+5 \times 10^{-3}$
(iii) The radius of a hydrogen atom is $2.5 \times 10^{-11} \mathrm{~m}$
Answer:
$\begin{aligned}
&\text { (i) } 8 \times 10^{4}+7 \times 10^{3}+6 \times 10^{2}+5 \times 10^{1}+2 \times 1+4 \times 10^{-2}+7 \times 10^{-4} \\
&=8 \times 10^{4}+7 \times 10^{3}+6 \times 10^{2}+5 \times 10^{1}+2 \times 1+4 \times 10^{-2}+7 \times 10^{-4} \\
&=8 \times 10000+7 \times 1000+6 \times 100+5 \times 10+2 \times 1+4 \times \frac{1}{100}+7 \times \frac{1}{10000} \\
&=80000+7000+600+50+2+\frac{4}{100}+\frac{7}{10000} \\
&=87652.0407
\end{aligned}$

$\begin{aligned}
&\text { (ii) } 5 \times 10^{3}+5 \times 10^{1}+5 \times 10^{-1}+5 \times 10^{-3} \\
&=5 \times 10^{3}+5 \times 10^{1}+5 \times 10^{-1}+5 \times 10^{-3} \\
&=5 \times 1000+5 \times 10+5 \times \frac{1}{10}+5 \times \frac{1}{1000} \\
&=5000+50+\frac{5}{10}+\frac{5}{1000}=5050.505
\end{aligned}$
(iii) The radius of a hydrogen atom is $2.510^{-11} \mathrm{~m}$
Radiys of a hydrogen atom $=2.5 \times 10^{-11} \mathrm{~m}$
$\begin{aligned}
&=2.5 \times \frac{1}{10^{11}} \mathrm{~m}=\frac{2.5}{10^{11}} \mathrm{~m} \\
&=0.00000000025 \mathrm{~m}
\end{aligned}$

Question $10 .$
Write the following numbers in scientific notation:
(i) 467800000000
Answer:
$467800000000=4.678 \times 10^{11}$
(ii) $0.000001972$
Answer:
$0.000001972=1.972 \times 10^{-6}$
(iii) $1642.398$
Answer:
$1642.398=1.642398 \times 10^{3}$
(iv) Earth's volume is about $1,083,000,000,000$ cubic kilometres
Answer:
$1,083,000,000,000$
Earth's volume $=1.083110 \times 10^{2}$ cubic kilometres

(v) If you fill a bucket with dirt, the portion of the whole Earth that is in the bucket will be $0.0000000000000000000000016 \mathrm{~kg}$
Answer:
Portion of earth in the bucket $=0.0000000000000000000000016 \mathrm{~kg}$ $=1.610 \times 10^{24} \mathrm{~kg}$.
 

Objective Type Questions
Question 11 .
By what number should $(-4)^{-1}$ be multiplied so that the product becomes $10^{-1}$ ?
(A) $\frac{2}{3}$
(B) $\frac{-2}{5}$
(C) $\frac{5}{2}$
(D) $\frac{-5}{2}$
Answer:
(B) $\frac{-2}{5}$

Question 12 .
$(-2)^{-3} \times(-2)^{-2}=$
(A) $\frac{-1}{32}$
(B) $\frac{1}{32}$
(C) 32
(D) $-32$
Answer:
(A) $\frac{-1}{32}$
 

Question $13 .$
Which is not correct?
(A) $\left(\frac{-1}{4}\right)^{2}=4^{-2}$
(B) $\left(\frac{-1}{4}\right)^{2}=\left(\frac{1}{2}\right)^{4}$
(C) $\left(\frac{-1}{4}\right)^{2}=16^{-1}$
(D) $-\left(\frac{1}{4}\right)^{2}=16^{-1}$
Answer:
$-\left(\frac{1}{4}\right)^{2}=16^{-1}$

Question $14 .$
If $\frac{10^{x}}{10^{-3}}=10^{9}$, then $\mathrm{x}$ is
(A) 4
(B) 5
(C) 6
(D) 7

Question $15 .$
$0.0000000002020$ in scientific form is
(A) $2.02 \times 10^{9}$
(B) $2.02 \times 10^{-9}$
(C) $2.02 \times 10^{-8}$
(D) $2.02 \times 10^{-10}$
Answer:
(D) $2.02 \times 10^{-10}$
Hint
$0.0000000002020$