Exercise 2.5 - Chapter 2 Real Numbers 9th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $2.5$
Question $1 .$
Write the following in the form of $5^{n}$ :
(i) 625
(ii) $\frac{1}{5}$
(iii) $\sqrt{5}$
(iv) $\sqrt{125}$
Solution:

(ii) $\frac{1}{5}=5^{-1}$
(iii) $\sqrt{5}=5^{\frac{1}{2}}$
(iv) $\sqrt{125}$
$=\sqrt{5^{3}}=\left(5^{3}\right)^{\frac{1}{2}}=5^{\frac{3}{2}}$

Question $2 .$
Write the following in the form of $4^{\mathrm{n}}$ :
(i) 16
(ii) 8
(iii) 32
Solution:
(i) $16=4^{2}$
(ii) $8=4^{1} \times 4^{\frac{1}{2}}=4^{1+\frac{1}{2}}=4^{\frac{3}{2}}$
(iii) $32=4 \times 4 \times 4^{\frac{1}{2}}=4^{2+\frac{1}{2}}=4^{\frac{5}{2}}$

Question 3 .
Find the value of
(i) $(49)^{\frac{1}{2}}$
(ii) $(243)^{\frac{2}{5}}$
(iii) $9^{\frac{-3}{2}}$
(iv) $\left(\frac{64}{125}\right)^{\frac{-2}{3}}$
Solution:
(i)$(49)^{\frac{1}{2}}=(7 \times 7)^{\frac{1}{2}}=7$
(ii) $(243)^{\frac{2}{5}}=\left(3^{5}\right)^{\frac{2}{5}}=3^{\5 \times \frac{2}{5}}=3^{2}=9$

(iii) $9^{\frac{-3}{2}}=\left(3^{2}\right)^{\frac{-3}{2}}=3^{\not \times \times \frac{-3}{2}}=3^{-3}=\frac{1}{3^{3}}=\frac{1}{27}$
(iv) $\left(\frac{64}{125}\right)^{\frac{-2}{3}}=\left(\frac{125}{64}\right)^{\frac{2}{3}}=\left(\left(\frac{5}{4}\right)^{3}\right)^{\frac{2}{3}}=\left(\frac{5}{4}\right)^{3 / \times \frac{2}{3}}=\left(\frac{5}{4}\right)^{2}=\frac{25}{16}$
 

Question $4 .$
Use a fractional index to write:
(i) $\sqrt{5}$
(ii) $\sqrt[2]{7}$
(iii) $(\sqrt[3]{49})^{5}$
((iv) $\left(\frac{1}{\sqrt[3]{100}}\right)^{7}$
Solution:
(i) $\sqrt{5}=5^{\frac{1}{2}}$
(ii) $\sqrt[2]{7}=7^{\frac{1}{2}}$

(iii) $(\sqrt[3]{49})^{5}=49^{\frac{5}{3}}$
(iv) $\left(\frac{1}{\sqrt[3]{100}}\right)^{7}=\left(\frac{1}{100^{\frac{1}{3}}}\right)^{7}=\left(\frac{1^{\frac{1}{3}}}{100^{\frac{1}{3}}}\right)^{7}=\left(\frac{1}{100}\right)^{\frac{1}{3}}$

Question 5 .
Find the $5^{\text {th }}$ root of
(i) 32
(ii) 243
(iii) 100000
(iv) $\frac{1024}{3125}$
Solution:
(i) $\sqrt[5]{32}=32^{\frac{1}{5}}=\left(2^{5}\right)^{\frac{1}{5}}=2^{8 \times \frac{1}{8}}=2$
(ii)$\sqrt[5]{243}=243^{\frac{1}{5}}=\left(3^{5}\right)^{\frac{1}{5}}=3^{\$ \times \frac{1}{3}}=3$
(iii) $\sqrt[5]{100000}=(100000)^{\frac{1}{5}}=\left(10^{8}\right)^{\frac{1}{\beta}}=10$
(iv) $\sqrt[5]{\frac{1024}{3125}}=\left(\frac{1024}{3125}\right)^{\frac{1}{5}}=\left(\left(\frac{4}{5}\right)^{\beta}\right)^{\frac{1}{8}}=\frac{4}{5}$