Exponents and powers in mathematics are used to express repeated multiplication of a number. An exponent (also known as a power) is written as a small number above and to the right of the base number, indicating how many times the base is multiplied by itself. For example:
(i) 2⁶
= 2 x 2 x 2 x 2 x 2 x 2
= 64
This means 2 is multiplied by itself 6 times.
(ii) 5²
= 5 x 5
= 25
Here, 5 is multiplied by itself, termed as ‘five squared’.
Special cases include any number raised to the power of zero, which always equals 1, e.g., 7⁰ = 1, and negative exponents representing the reciprocal, e.g., 2⁻³ = 1/(2³) = 1/8. Understanding these rules helps simplify expressions and solve problems involving powers.”
NCERT Solutions for Class 7 Maths Exercise 11.1 Chapter 11 Exponents and Powers
1. Find the value of:
(i) 2⁶
= 2 x 2 x 2 x 2 x 2 x 2 (Multiplying 2 six times because the exponent is 6)
= 64 (2 multiplied by itself 6 times equals 64)
(ii) 9³
= 9 x 9 x 9 (Multiplying 9 three times because the exponent is 3)
= 729 (9 multiplied by itself 3 times equals 729)
(iii) 11²
= 11 x 11 (Multiplying 11 two times because the exponent is 2)
= 121 (11 multiplied by itself 2 times equals 121)
(iv) 5⁴
= 5 x 5 x 5 x 5 (Multiplying 5 four times because the exponent is 4)
= 625 (5 multiplied by itself 4 times equals 625)
2. Express the following in exponential form:
(i) 6 × 6 × 6 × 6
= 6⁴ (Four 6s multiplied together, so we use the exponent 4)
(ii) t × t
= t² (Two ts multiplied together, so we use the exponent 2)
(iii) b × b × b × b
= b⁴ (Four bs multiplied together, so we use the exponent 4)
(iv) 5 × 5 × 7 × 7 × 7
= 5² x 7³ (Two 5s and three 7s multiplied, so we use exponents 2 and 3 respectively)
(v) 2 × 2 × a × a
= 2² x a² (Two 2s and two as multiplied, so we use exponents 2 for each)
(vi) a × a × a × c × c × c × c × d
= a³ x c⁴ x d (Three as, four cs, and one d multiplied, so we use exponents 3, 4, and 1 respectively)
3. Express each of the following numbers using exponential notation:
(i) 512
= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 (512 is the product of nine 2s)
= 2⁹ (So, 512 is expressed as 2 raised to the power of 9)
(ii) 343
= 7 x 7 x 7 (343 is the product of three 7s)
= 7³ (So, 343 is expressed as 7 raised to the power of 3)
(iii) 729
= 3 x 3 x 3 x 3 x 3 x 3 (729 is the product of six 3s)
= 3⁶ (So, 729 is expressed as 3 raised to the power of 6)
(iv) 3125
= 5 x 5 x 5 x 5 x 5 (3125 is the product of five 5s)
= 5⁵ (So, 3125 is expressed as 5 raised to the power of 5)
4. Identify the greater number, wherever possible, in each of the following?
(i) 4³ or 3⁴
4³ = 4 x 4 x 4 = 64; 3⁴ = 3 x 3 x 3 x 3 = 81 (Comparing the products)
Greater: 3⁴ (81 is greater than 64)
(ii) 5³ or 3⁵
5³ = 5 x 5 x 5 = 125; 3⁵ = 3 x 3 x 3 x 3 x 3 = 243 (Comparing the products)
Greater: 3⁵ (243 is greater than 125)
(iii) 2⁸ or 8²
2⁸ = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256; 8² = 8 x 8 = 64 (Comparing the products)
Greater: 2⁸ (256 is greater than 64)
(iv) 100² or 2¹⁰⁰
100² = 100 x 100 = 10,000; 2¹⁰⁰ is a much larger number (Note that 2¹⁰⁰ is a very large number, much larger than 10,000)
Greater: 2¹⁰⁰ (2¹⁰⁰ is greater than 10,000)
(v) 2¹⁰ or 10²
2¹⁰ = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1024; 10² = 10 x 10 = 100 (Comparing the products)
Greater: 2¹⁰ (1024 is greater than 100)
5. Express each of the following as product of powers of their prime factors:
(i) 648
= 2 x 2 x 2 x 3 x 3 x 3 x 3 (Breaking 648 into its prime factors)
= 2³ x 3⁴ (Grouping the prime factors into powers)
(ii) 405
= 3 x 3 x 3 x 3 x 5 (Breaking 405 into its prime factors)
= 3⁴ x 5 (Grouping the prime factors into powers)
(iii) 540
= 2 x 2 x 3 x 3 x 3 x 5 (Breaking 540 into its prime factors)
= 2² x 3³ x 5 (Grouping the prime factors into powers)
(iv) 3,600
= 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 (Breaking 3,600 into its prime factors)
= 2⁴ x 3² x 5² (Grouping the prime factors into powers)
6. Simplify:
(i) 2 × 10³
= 2 x 1000 (10³ means 10 multiplied by itself 3 times, which is 1000)
= 2000
(ii) 7² × 2²
= 49 x 4 (7² is 49 and 2² is 4)
= 196
(iii) 2³ × 5
= 8 x 5 (2³ means 2 multiplied by itself 3 times, which is 8)
= 40
(iv) 3 × 4⁴
= 3 x 256 (4⁴ is 4 multiplied by itself 4 times, which is 256)
= 768
(v) 0 × 10²
= 0 x 100 (Any number multiplied by 0 is 0)
= 0
(vi) 5² × 3³
= 25 x 27 (5² is 25 and 3³ is 27)
= 675
(vii) 2⁴ × 3²
= 16 x 9 (2⁴ is 16 and 3² is 9)
= 144
(viii) 3² × 10⁴
= 9 x 10000 (3² is 9 and 10⁴ is 10,000)
= 90000
7. Simplify:
(i) (– 4)³
= –4 x –4 x –4 (–4 multiplied by itself 3 times)
= –64 (Negative number raised to an odd power is negative)
(ii) (–3) × (–2)³
= –3 x (–8) (–2³ is –2 multiplied by itself 3 times, which is –8)
= 24 (Negative times negative is positive)
(iii) (–3)² × (–5)²
= 9 x 25 (Negative number squared is positive)
= 225
(iv) (–2)³ × (–10)³
= (–8) x (–1000) (Negative number raised to an odd power is negative)
= 8000 (Negative times negative is positive)
8. Compare the following numbers:
(i) 2.7 × 10¹² ; 1.5 × 10⁸
= 2,700,000,000,000 ; 150,000,000 (Expanding the numbers)
Greater: 2.7 x 10¹² (2,700,000,000,000 is greater than 150,000,000)
(ii) 4 x 10¹⁴ ; 3 x 10¹⁷
= 400,000,000,000,000 ; 300,000,000,000,000,000 (Expanding the numbers)
Greater: 3 x 10¹⁷ (300,000,000,000,000,000 is greater than 400,000,000,000,000)
