# RD Sharma Solutions Class 7 Exponents Exercise 6.1

## RD Sharma Solutions Class 7 Chapter 6 Exercise 6.1

### RD Sharma Class 7 Solutions Chapter 6 Ex 6.1 PDF Free Download

#### Exercise 6.1

Q1. Find the values of each of the following :

(i) $13^{2}$

(ii) $7^{3}$

(iii) $3^{4}$

Sol:

(i) $13^{2}$ = 13$\times$13

= 169

(ii) $7^{3}$ = 7$\times$7$\times$7

= 4

(iii) $3^{4}$ = 3$\times$3$\times$3$\times$3

= 81

Q2. Find the value of each of the following :

(i) $(-7)^{2}$

(ii) $(-3)^{4}$

(iii) $(-5)^{5}$

Sol:

We know that if ‘a’ is a natural number, then

$(-a)^{even\;number}$ = positive number

$(-a)^{odd\;number}$ = negative number

We have,

(i) $(-7)^{2}$ = (-7) $\times$(-7)

= 49

(ii) $(-3)^{4}$ = (-3) $\times$(-3) $\times$(-3) $\times$(-3)

= 81

(iii) $(-5)^{5}$ = (-5) $\times$(-5) $\times$(-5) $\times$(-5) $\times$(-5)

= -3125

Q3. Simply :

(i) $3\times 10^{2}$

(ii) $2^{2}\times 5^{3}$

(iii) $3^{3}\times 5^{2}$

Sol:

(i) $3\times 10^{2}$ = 3$\times$10$\times$10

= 3$\times$100

= 300

(ii) $2^{2}\times 5^{3}$ = 2$\times$2$\times$5$\times$5$\times$5

= 4$\times$125

= 500

(iii) $3^{3}\times 5^{2}$ = 3$\times$3$\times$3$\times$5$\times$5

= 27$\times$25

= 675

Q4. Simply :

(i) $3^{2}\times 10^{4}$

(ii) $2^{4}\times 3^{2}$

(iii) $5^{2}\times 3^{4}$

Sol:

(i) $3^{2}\times 10^{4}$ = 3$\times$3$\times$10$\times$10$\times$10$\times$10

= 9$\times$10000

= 90000

(ii) $2^{4}\times 3^{2}$ = 2$\times$2$\times$2$\times$2$\times$3$\times$3

= 16$\times$9

= 144

(iii) $5^{2}\times 3^{4}$ = 5$\times$5$\times$3$\times$3$\times$3$\times$3

= 25$\times$81

= 2025

Q5. Simply :

(i) $(-2)\times (-3)^{3}$

(ii) $(-3)^{2}\times (-5)^{3}$

(iii) $(-2)^{5}\times (-10)^{2}$

Sol:

(i) $(-2)\times (-3)^{3}$ = (-2) $\times$(-3) $\times$(-3) $\times$(-3)

= (-2) $\times$(-27)

= 54

(ii) $(-3)^{2}\times (-5)^{3}$ = (-3) $\times$(-3) $\times$(-5) $\times$(-5) $\times$(-5)

= 9$\times$(-125)

= -1125

(iii) $(-2)^{5}\times (-10)^{2}$ = (-2) $\times$(-2) $\times$(-2) $\times$(-2) $\times$(-2) $\times$(-10) $\times$(-10)

= (-32) $\times$ 100

= -3200

Q6. Simply :

(i) $(\frac{3}{4})^{2}$

(ii) $(\frac{-2}{3})^{4}$

(iii) $(\frac{-4}{5})^{5}$

Sol:

(i) $(\frac{3}{4})^{2}$ = $\frac{3\times 3}{4\times 4}$

= $\frac{9}{16}$

(ii) $(\frac{-2}{3})^{4}$ = $\frac{(-2)\times (-2)\times (-2)\times (-2)}{3\times 3\times 3\times 3}$

= $\frac{16}{81}$

(iii) $(\frac{-4}{5})^{5}$ =$\frac{(-4)\times (-4)\times (-4)\times (-4)\times (-4)}{5\times 5\times 5\times 5\times 5}$

= $\frac{-1024}{3125}$

Q7. Identify the greater number in each of the following

(i) $2^{5}$ or $5^{2}$

(ii) $3^{4}$ or $4^{3}$

(iii) $3^{5}$ or $5^{3}$

Sol:

(i) $2^{5}$ or $5^{2}$

= $2^{5}$ = 2$\times$2$\times$2$\times$2$\times$2

= 32

= $5^{2}$ = 5$\times$5

= 25

Therefore, $2^{5}$ $5^{2}$

(ii) $3^{4}$ or $4^{3}$

= $3^{4}$ = 3$\times$3$\times$3$\times$3

= 81

= $4^{3}$ = 4$\times$4$\times$4

= 64

Therefore, $3^{4}$ $4^{3}$

(iii) $3^{5}$ or $5^{3}$

= $3^{5}$ = 3$\times$3$\times$3$\times$3$\times$3

= 243

= $5^{3}$ = 5$\times$5$\times$5

= 125

Therefore, $3^{5}$ $5^{3}$

Q8. Express each of the following in exponential form

(i) (-5) $\times$(-5) $\times$(-5)

(ii) $(\frac{-5}{7}\times \frac{-5}{7}\times \frac{-5}{7}\times \frac{-5}{7})$

(iii) $(\frac{4}{3}\times \frac{4}{3}\times \frac{4}{3}\times \frac{4}{3}\times \frac{4}{3})$

Sol:

(i) (-5) $\times$(-5) $\times$(-5) = $(-5)^{3}$

(ii) $(\frac{-5}{7}\times \frac{-5}{7}\times \frac{-5}{7}\times \frac{-5}{7})$ = $(\frac{-5}{7})^{4}$

(iii) $(\frac{4}{3}\times \frac{4}{3}\times \frac{4}{3}\times \frac{4}{3}\times \frac{4}{3})$ = $(\frac{4}{3})^{5}$

Q9. Express each of the following in exponential form

(i) x $\times$x $\times$x $\times$x $\times$a $\times$a $\times$b $\times$b $\times$b

(ii) (-2) $\times$(-2) $\times$(-2) $\times$(-2) $\times$a$\times$a$\times$a

(iii) $(\frac{-2}{3})\times (\frac{-2}{3})\times x\times x\times x$

Sol:

(i) x $\times$x $\times$x $\times$x $\times$a $\times$a $\times$b $\times$b $\times$b  = $x^{4}a^{2}b^{3}$

(ii) (-2) $\times$(-2) $\times$(-2) $\times$(-2) $\times$a$\times$a$\times$a = $(-2)^{4}a^{3}$

(iii) $(\frac{-2}{3})\times (\frac{-2}{3})\times x\times x\times x$ = $(\frac{-2}{3})^{2}x^{3}$

Q10. Express each of the following numbers in exponential form

(i) 512

(ii) 625

(iii) 729

Sol:

(i) 512 = $2^{9}$

(iii) 625 = $5^{4}$

(iii) 729 = $3^{6}$

Q11. Express each of the following numbers as a product of powers of their prime factors

(i) 36

(ii) 675

(iii) 392

Sol:

(i) 36 = $2\times 2\times 3\times 3$

= $2^{2}\times 3^{2}$

(ii) 675 = $3\times 3\times 3\times 5\times 5$

= $3^{3}\times 5^{2}$

(iii) 392 = $2\times 2\times 2\times 7\times 7$

= $2^{3}\times 7^{2}$

Q12. Express each of the following numbers as a product of powers of their prime factors

(i) 450

(ii) 2800

(iii) 24000

Sol:

(i) 450 = $2\times 3\times 3\times 5\times 5$

= $2\times 3^{2}\times 5^{2}$

(ii) 2800 = $2\times 2\times 2\times 2\times 5\times 5\times 7$

= $2^{4}\times 5^{2}\times 7$

(iii) 24000 = $2\times 2\times 2\times 2\times 2\times 2\times 3\times 5\times 5\times 5$

= $2^{5}\times 3\times 5^{3}$

Q13. Express each of the following as a rational number of the form $\frac{p}{q}$

(i) $(\frac{3}{7})^{2}$

(ii) $(\frac{7}{9})^{3}$

(iii) $(\frac{-2}{3})^{4}$

Sol:

(i) $(\frac{3}{7})^{2}$ = $\frac{3\times 3}{7\times 7}$

= $\frac{9}{49}$

(ii) $(\frac{7}{9})^{3}$ = $\frac{7\times 7\times 7}{9\times 9\times 9}$

= $\frac{343}{729}$

(iii) $(\frac{-2}{3})^{4}$ = $\frac{(-2)\times (-2)\times (-2)\times (-2)}{3\times 3\times 3\times 3}$

= $\frac{16}{81}$

Q14. Express each of the following rational numbers in power notation

(i) $\frac{49}{64}$

(ii) $-\frac{64}{125}$

(iii) $-\frac{1}{216}$

Sol:

(i) $\frac{49}{64}$ = $(\frac{7}{8})^{2}$

Because $7^{2}$ = 49 and $8^{2}$ = 64

(ii) $-\frac{64}{125}$ = $(-\frac{4}{5})^{3}$

Because $4^{3}$ = 64 and $5^{3}$ = 125

(iii) $-\frac{1}{216}$ = $(-\frac{1}{6})^{3}$

Because $1^{3}$ = 1 and $6^{3}$ = 216

Q15. Find the value of the following

(i) $(\frac{-1}{2})^{2}\times 2^{3}\times (\frac{3}{4})^{2}$

(ii) $(\frac{-3}{5})^{4}\times (\frac{4}{9})^{4}\times (\frac{-15}{18})^{2}$

Sol:

(i) $(\frac{-1}{2})^{2}\times 2^{3}\times (\frac{3}{4})^{2}$ = $\frac{1}{2}\times 8\times \frac{9}{16}$

= $\frac{9}{8}$

(ii) $(\frac{-3}{5})^{4}\times (\frac{4}{9})^{4}\times (\frac{-15}{18})^{2}$ = $\frac{81}{625}\times \frac{256}{6561}\times \frac{225}{324}$ = $\frac{64}{18225}$

Q16. If a= 2 and b = 3, the find the values of each of the followimg

(i) $(a+b)^{a}$

(ii) $(ab)^{b}$

(iii) $(\frac{b}{a})^{b}$

(iv) $(\frac{a}{b}+\frac{b}{a})^{a}$

Sol:

(i) $(a+b)^{a}$ = $(2+3)^{2}$

= $(5)^{2}$

= 25

(ii) $(ab)^{b}$ = $(2\times 3)^{3}$

= $(6)^{3}$

= 216

(iii) $(\frac{b}{a})^{b}$ = $(\frac{3}{2})^{3}$

= $\frac{27}{8}$

(iv) $(\frac{a}{b}+\frac{b}{a})^{a}$ = $(\frac{2}{3}+\frac{3}{2})^{2}$

= $\frac{169}{36}$

#### Practise This Question

Simplify (67)2×(76)2