Page 87 - Math_Genius_V1.0_C7_Flipbook

D:\Surender Prajapati\CBSE_ICSE_Book_New\CBSE\Grade-7\Math_Genius-7\Open_File\04_Chapter\04_Chapter
\ 15-Nov-2024 Surender Prajapati Proof-6 Reader’s Sign _______________________ Date __________

2 4
Thus, (3 ) = 3 2 × 4
Let a be a non-zero integer then,

n
3 2
3
3
m
(a ) = a × a = a 3 + 3 (Q a × a = a m + n )
6
= a = a 3 × 2 (6 is the product of 3 and 2).
3 2
Thus, (a ) = a 3 × 2
In general, if a is any non-zero integer and m, n are whole numbers, then, (a ) = a mn .
m n
It is called the III law of exponents.

Example 11: Simplify the following in exponential form:

(a) (6 ) (b) (–5 )
3 7
4 3
2 3
3 4
(c) (11 ) (d) (–p )
mn
m n
Solution: (a) (6 ) = 6 4 × 3 = 6 12 (Q (a ) = a )
4 3
m n
mn
(b) (–5 ) = (–5) 3 × 7 = (–5) 21 (Q (a ) = a )
3 7
2 3
mn
(c) (11 ) = 11 2 × 3 = 11 6 (Q (a ) = a )
m n
m n
mn
3 4
(d) (–p ) = (–p) 3 × 4 = (–p) 12 (Q (a ) = a )
Quick Check
51 3
3 51
1. Which is greater: (–1 ) or (1 ) ?
5
2. Fill the possible whole number in the blank box to make the expression true. (3 ) = 3 20
Law IV: Multiplying Powers with the Same Exponents

3
Let us simplify 2 × 3 = (2 × 2 × 2) × (3 × 3 × 3)
3
= (2 × 3) × (2 × 3) × (2 × 3)
3
= 6 × 6 × 6 = 6 or (2 × 3) 3
Let us take another example,

4
4 × (–3) = (4 × 4 × 4 × 4) × (–3) × (–3) × (–3) × (–3)
4
= [4 × (–3)] × [4 × (–3)] × [4 × (–3)] × [4 × (–3)]
4
= (–12) × (–12) × (–12) × (–12) = (–12) = [4 × (–3)] 4
4
4
Thus, 4 × (–3) = (–12) = [4 × (–3)] 4
4
2
2
Also, take 2 × p = (2 × 2) × (p × p) = (2 × p) × (2 × p)
= (2 × p) = (2p) 2
2
4
4
Similarly, a × b = (a × a × a × a) × (b × b × b × b)
= (a × b) × (a × b) × (a × b) × (a × b)
4
= (a × b) = (ab) 4
85 Exponents and Powers