Exercise 1.5 - Chapter 1 - Number Systems - 8th Maths Guide Samacheer Kalvi Solutions

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Question 1.
Fill in the blanks
(i) The ones digits in the cube of 73 is
Answer:
7
(ii) The maximum number of digits in the cube of a two digit number is
Answer:
6
(iii) The smallest number to be added to 3333 to make it a perfect cube is
Answer:
42
(iv) The cube root of $540 \times 50$ is
Answer:
30
(v) The cube root of $0.000004913$ is
Answer:

0.017

Question $2 .$
Say True or False.
(i) The cube of 24 ends with the digit $4 .$
Answer:
True
(ii) Subtracting 103 from 1729 gives 93 .
Answer:
True
(iii) The cube of $0.0012$ is $0.000001728$.
Answer:
False
(iv) 79570 is not a perfect cube.
Answer:
True
(v) The cube root of 250047 is 63 .
Answer:

True

Question $3 .$
Show that 1944 is not a perfect cube.
Answer:

$\begin{aligned}
&1944=2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \\
&=2 \times 2 \times 2 \times \overline{3 \times 3 \times 3} \times 3 \times 3 \\
&=2^{3} \times 3^{3} \times 3 \times 3
\end{aligned}$
There are two triplets to make further triplets we need one more 3 .
$\therefore 1944$ is not a perfect cube.

Question $4 .$
Find the smallest number by which 10985 should be divided so that the quotient is a perfect cube.
Answer:

We have $10985=5 \times 13 \times 13 \times 13$
$=5 \times 13 \times 13 \times 13$
Here we have a triplet of 13 and we are left over with $5 .$ If we divide 10985 by 5 , the new number will be a perfect cube. $\therefore$ The required number is 5 .

Question 5.

Find the smallest number by which 200 should be multiplied to make it a perfect cube.

Answer:

We find $200=\overline{2 \times 2 \times 2} \times 5 \times 5$
Grouping the prime factors of 200 as triplets, we are left with $5 \times 5$ We need one more 5 to make it a perfect cube.
So to make 200 a perfect cube multiply both sides by 5 .
$\begin{aligned}
&200 \times 5=(\overline{2 \times 2 \times 2} \times 5 \times 5) \times 5 \\
&1000=2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5 \times 5
\end{aligned}$
Now 1000 is a perfect cube.
$\therefore$ The required number is 5 .

Question $6 .$
Find the cube root $24 \times 36 \times 80 \times 25$.
Answer:

Question $7 .$
Find the cube root of 729 and 6859 prime factorisation.

Answer:

Question 8 .
What is the square root of cube root of $46656 ?$
Answer:

We have to find out $\sqrt{(\sqrt[3]{46656})}$
First we will find $\sqrt[3]{46656}$
$\begin{aligned}
&\sqrt[3]{46656}=(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3)^{\frac{1}{3}} \\
&\sqrt[3]{46656}=2 \times 2 \times 3 \times 3 \\
&\sqrt[3]{46656}=2^{2} \times 3^{2}=36
\end{aligned}$
Now $\sqrt{(\sqrt[3]{46656})}=\sqrt{36}=\sqrt{2^{2} \times 3^{2}}=2 \times 3=6$
$\therefore$ The required number is 6 .

Question $9 .$
If the cube of a squared number is 729 , find the square root of that number.
Answer:
$729=\overline{3 \times 3 \times 3} \times \overline{3 \times 3 \times 3}$
$(729)^{1 / 3}=3 \times 3=9$
$\therefore$ The cube of 9 is 729 .
$9=3 \times 3$ [ie 3 is squared to get 9$]$

We have to find out $\sqrt{3}$,
$\sqrt{3}=1.732$
 

Question $10 .$
Find two smallest perfect square numbers which when multiplied together gives a perfect cube number.
Answer:
Consider the numbers $2^{2}$ and $4^{2}$
The numbers are 4 and 16 .
Their procluct $4 \times 16=64$
$64=4 \times 4 \times 4$
$\therefore$ The required square numbers are 4 and 16