Chapter 13 Exponents and Powers

Chapter 13 Exponents and Powers

Very Short Answer Type

Question 1.
Express 343 as a power of 7.
Solution:
We have 343 = 7 × 7 × 7 = 7³
Thus, 343 = 7³

Question 2.
Which is greater 3² or 2³?
Solution:
We have 3² = 3 × 3 = 9
2³ = 2 × 2 × 2 = 8
Since 9 > 8
Thus, 3² > 2³

Question 3.
Express the following number as a powers of prime factors:
(i) 144
(ii) 225
Solution:
(i) We have
144 = 2 × 2 × 2 × 2 × 3 × 3 = 2⁴ × 3²
Thus, 144 = 2⁴ × 3²

(ii) We have
225 = 3 × 3 × 5 × 5 = 3² × 5²
Thus, 225 = 3² × 5²

Question 4.
Find the value of:

Question 5.
Express the following in exponential form:
(i) 5 × 5 × 5 × 5 × 5
(ii) 4 × 4 × 4 × 5 × 5 × 5
(iii) (-1) × (-1) × (-1) × (-1) × (-1)
(iv) a × a × a × b × c × c × c × d × d
Solution:
(i) 5 × 5 × 5 × 5 × 5 = (5)⁵
(ii) 4 × 4 × 4 × 5 × 5 × 5 = 4³ × 5³
(iii) (-1) × (-1) × (-1) × (-1) × (-1) = (-1)⁵
(iv) a × a × a × b × c × c × c × d × d = a³b¹c³d²

Question 6.
Express each of the following as product of powers of their prime factors:
(i) 405
(ii) 504
(iii) 500
Solution:
(i) We have
405 = 3 × 3 × 3 × 3 × 5 = 3⁴ × 5¹
Thus, 405 = 3⁴ × 5¹

(ii) We have
504 = 2 × 2 × 2 × 3 × 3 × 7 = 2³ × 3² × 7¹
Thus, 504 = 2³ × 3² × 7¹

(iii) We have
500 = 2 × 2 × 5 × 5 × 5 = 2² × 5³
Thus, 500 = 2² × 5³

Question 7.
Simplify the following and write in exponential form:
(i) (5²)³
(ii) (2³)³
(iii) (a^b)^c
(iv) [(5)²]²
Solution:
(i) (5²)³ = 5^2×3 = 5⁶
(ii) (2³)³ = 2^3×3 = 2⁹
(iii) (a^b)^c = a^b×c = a^bc
(iv) [(5)²]² = 5^2×2 = 5⁴

Question 8.
Verify the following:

Solution:

Question 9.
Simplify:

Solution:

Question 10.
Simplify and write in exponential form:

Solution:

Short Answer Type

Question 11.
Express each of the following as a product of prime factors is the exponential form:
(i) 729 × 125
(ii) 384 × 147
Solution:
(i) 729 × 125 = 3 × 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5 = 3⁶ × 5³
Thus, 729 × 125 = 3⁶ × 5³

(ii) 384 × 147 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7 = 2⁷ × 3² × 7²
Thus, 384 × 147 = 2⁷ × 3² × 7²

Question 12.
Simplify the following:
(i) 10³ × 9⁰ + 3³ × 2 + 7⁰
(ii) 6³ × 7⁰ + (-3)⁴ – 9⁰
Solution:
(i) 10³ × 9⁰ + 3³ × 2 + 7⁰
= 1000 + 54 + 1
= 1055
(ii) 6³ × 7⁰ + (-3)⁴ – 9⁰
= 216 × 1 + 81 – 1
= 216 + 80
= 296

Question 13.
Write the following in expanded form:
(i) 70,824
(ii) 1,69,835
Solution:
(i) 70,824
= 7 × 10000 + 0 × 1000 + 8 × 100 + 2 × 10 + 4 × 10⁰
= 7 × 10⁴ + 8 × 10² + 2 × 10¹ + 4 × 10⁰
(ii) 1,69,835
= 1 × 100000 + 6 × 10000 + 9 × 1000 + 8 × 100 + 3 × 10 + 5 × 10⁰
= 1 × 10⁵ + 6 × 10⁴ + 9 × 10³ + 8 × 10² + 3 × 10¹ + 5 × 10⁰

Question 14.
Find the number from each of the expanded form:
(i) 7 × 10⁸ + 3 × 10⁵ + 7 × 10² + 6 × 10¹ + 9
(ii) 4 × 10⁷ + 6 × 10³ + 5
Solution:
(i) 7 × 10⁸ + 3 × 10⁵ + 7 × 10² + 6 × 10¹ + 9
= 7 × 100000000 + 3 × 100000 + 7 × 100 + 6 × 10 + 9
= 700000000 + 300000 + 700 + 60 + 9
= 700300769
(ii) 4 × 10⁷ + 6 × 10³ + 5
= 4 × 10000000 + 6 × 1000 + 5
= 40000000 + 6000 + 5
= 40006005

Question 15.
Find the value of k in each of the following:

Solution:

Question 16.
Find the value of
(a) 3⁰ ÷ 4⁰
(b) (8⁰ – 2⁰) ÷ (8⁰ + 2⁰)
(c) (2⁰ + 3⁰ + 4⁰) – (4⁰ – 3⁰ – 2⁰)
Solution:
(a) We have 3⁰ ÷ 4⁰ = 1 ÷ 1 = 1 [∵ a⁰ = 1]
(6) (8⁰ – 2⁰) ÷ (8⁰ + 2⁰) = (1 – 1) ÷ (1 + 1) = 0 ÷ 2 = 0
(c) (2⁰ + 3⁰ + 4⁰) – (4⁰ – 3⁰ – 2⁰)
= (1 + 1 + 1) – (1 – 1 – 1) [∵ a⁰ = 1]
= 3 – 1
= 2

Question 17.
Express the following in standard form:
(i) 8,19,00,000
(ii) 5,94,00,00,00,000
(iii) 6892.25
Solution:
(i) 8,19,00,000 = 8.19 × 10⁷
(ii) 5,94,00,00,00,000 = 5.94 × 10¹¹
(iii) 6892.25 = 6.89225 × 10³

Question 18.
Evaluate:

Solution:

Higher Order Thinking Skills (HOTS) Type

Question 19.
Find the value of x, if

Solution:

Question 20.

Solution: