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

= \(\frac{169}{36}\)