RBSE Solutions for Class 8 Maths Chapter 12 Exponents and Powers Ex 12.1

Rajasthan Board RBSE Solutions for Class 8 Maths Chapter 12 Exponents and Powers Ex 12.1 Textbook Exercise Questions and Answers.

RBSE Class 8 Maths Solutions Chapter 12 Exponents and Powers Ex 12.1

Question 1.
Evaluate:
(i) 3^-2
Answer:
3^-2 = \(\frac{1}{3^{2}}=\frac{1}{3 \times 3}=\frac{1}{9}\)

(ii) (- 4)^-2
Answer:
(- 4)^-2 = \(\frac{1}{(-4)^{2}}=\frac{1}{(-1)^{2} \times(4)^{2}}\)
= \(\frac{1}{1 \times 16}=\frac{1}{16}\)

(iii) \(\left(\frac{1}{2}\right)^{-5}\)
Answer:
\(\left(\frac{1}{2}\right)^{-5}=\frac{1}{2^{-5}}\) = 2⁵ = 32

Question 2.
Simplify and express the result in power notation with positive exponent :
(i) (-4)⁵ ÷ (-4)⁸
Answer:
(-4)⁵ ÷ (-4)⁸ = \(\frac{(-4)^{5}}{(-4)^{8}}=\frac{1}{(-4)^{8-5}}\)
= \(\frac{1}{(-4)^{3}}\)

(ii) \(\left(\frac{1}{2^{3}}\right)^{2}\)
Answer:
\(\left(\frac{1}{2^{3}}\right)^{2}=\frac{1}{2^{3 \times 2}}=\frac{1}{2^{6}}\)

(iii) (-3)⁴ × \(\left(\frac{5}{3}\right)^{4}\)
Answer:
(-3)⁴ × \(\left(\frac{5}{3}\right)^{4}\) = (-1)⁴ × 3⁴ × \(\frac{5^{4}}{3^{4}}\)
= 1 × 3^4-4 × 5⁴ = 3⁰ × 5⁴
= 1 × 5⁴ = 5⁴

(iv) (3^-7 ÷ 3¹⁰) × 3^-5
Answer:
(3^-7 ÷ 3¹⁰) × 3^-5 = (3^-7 × \(\frac{1}{3^{-10}}\)) × 3^-5
= \(\frac{1}{3^{-10+7}}\) × 3^-5 = \(\frac{1}{3^{-3}}\) × 3^-5
= \(\frac{1}{3^{-3+5}}=\frac{1}{3^{2}}\)

(v) 2^-3 × (-7)^-3
Answer:
2^-3 × (-7)^-3 = \(\frac{1}{2^{3}} \times \frac{1}{(-7)^{3}}\)
= \(\frac{1}{(2 \times-7)^{3}}=\frac{1}{(-14)^{3}}\)

Question 3.
Find the value of:
(i) (3° + 4^-1) × 2²
Answer:
(3° + 4^-1) × 2² = (1 + \(\frac{1}{4}\)) × 4
= \(\frac{4+1}{4}\) × 4 = \(\frac{5}{4}\) ×4 = 5

(ii) (2^-1 × 4^-1) ÷ 2^-2
Answer:
(2^-1 × 4^-1) ÷ 2^-2 = \(\left(\frac{1}{2} \times \frac{1}{4}\right) \div \frac{1}{2^{2}}=\frac{1}{8} \div \frac{1}{4} = \frac{1}{8} \times \frac{4}{1}=\frac{1}{2}\)

(iii) \(\left(\frac{1}{2}\right)^{-2}+\left(\frac{1}{3}\right)^{-2}+\left(\frac{1}{4}\right)^{-2}\)
Answer:
\(\left(\frac{1}{2}\right)^{-2}+\left(\frac{1}{3}\right)^{-2}+\left(\frac{1}{4}\right)^{-2}\) = 2² + 3² + 4²
= 4 + 9 + 16 = 29

(iv) (3^-1 + 4^-1 + 5^-1)⁰
Answer:
(3^-1 + 4^-1 + 5^-1)⁰ = 1 [∵ a⁰ = 1]

(v) \(\left\{\left(\frac{-2}{3}\right)^{-2}\right\}^{2}\)
Answer:
\(\left\{\left(\frac{-2}{3}\right)^{-2}\right\}^{2}=\left\{\left(-\frac{3}{2}\right)^{2}\right\}^{2}=\left(\frac{9}{4}\right)^{2}=\frac{81}{16}\)

Question 4.
Evaluate
(i) \(\frac{8^{-1} \times 5^{3}}{2^{-4}}\)
Answer:
\(\frac{8^{-1} \times 5^{3}}{2^{-4}} = \frac{1}{8}\) × 5³ × 2⁴
= \(\frac{1}{8}\) × 125 × 16
= 125 × 2 = 250

(ii) (5^-1 × 2^-1) × 6^-1
Answer:
(5^-1 × 2^-1) × 6^-1 = \(\left(\frac{1}{5} \times \frac{1}{2}\right) \times \frac{1}{6}\)
= \(\frac{1}{10} \times \frac{1}{6}=\frac{1}{60}\)

Question 5.
Find the value of m for which
5^m ÷ 5^-3 = 5⁵
Answer:
Given, 5^m ÷ 5^-3 = 5⁵
or 5^m ÷ \(\frac{1}{5^{3}}\) = 5⁵
or 5^m × 5³ = 5⁵
or 5^m+3 = 5⁵
[∵ If a^m = aⁿ then m = n]
or m = 5 - 3
m = 2

Question 6.
Evaluate:
(i) \(\left\{\left(\frac{1}{3}\right)^{-1}-\left(\frac{1}{4}\right)^{-1}\right\}^{-1}\)
Answer:
\(\left\{\left(\frac{1}{3}\right)^{-1}-\left(\frac{1}{4}\right)^{-1}\right\}^{-1}\) = (3¹ - 4¹) = (-1)^-1
= \(\frac{1}{(-1)^{1}}=\frac{1}{-1}\) = -1

(ii) \(\left(\frac{5}{8}\right)^{-7} \times\left(\frac{8}{5}\right)^{-4}\)
Answer:
\(\left(\frac{5}{8}\right)^{-7} \times\left(\frac{8}{5}\right)^{-4}=\left(\frac{8}{5}\right)^{7} \times\left(\frac{5}{8}\right)^{4}\)
= \(\frac{8^{7}}{5^{7}} \times \frac{5^{4}}{8^{4}}=\frac{8^{7-4}}{5^{7-4}}\)
= \(\frac{8^{3}}{5^{3}}=\frac{512}{125}\)

Question 7.
Simplify:
(i) \(\frac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}}\) (t ≠ 0)
Answer:

(ii) \(\frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}\)
Answer: