EXPONENTS AND POWERS 251a × a × b × b × b × b can be expressed as a^2 b^4 (read as a
squared into braised to the power of 4).
EXAMPLE 1 Express 256 as a power 2.
SOLUTION We have 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2.
So we can say that 256 = 2^8
EXAMPLE 2 Which one is greater 2^3 or 3^2?
SOLUTION We have, 2^3 = 2 × 2 × 2 = 8 and 3^2 = 3 × 3 = 9.
Since 9 > 8, so, 3^2 is greater than 2^3
EXAMPLE 3 Which one is greater 8^2 or 2^8?
SOLUTION 82 = 8 × 8 = 64
28 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256
Clearly, 28 >8^2
EXAMPLE 4 Expanda^3 b^2 ,a^2 b^3 ,b^2 a^3 ,b^3 a^2. Are they all same?
SOLUTION a^3 b^2 =a^3 ×b^2
=(a×a×a) × (b×b)
=a×a×a×b×b
a^2 b^3 =a^2 × b^3
=a × a×b×b×b
b^2 a^3 =b^2 × a^3
=b × b×a×a × a
b^3 a^2 =b^3 × a^2
=b × b×b×a×a
Note that in the case of terms a^3 b^2 and a^2 b^3 the powers of a and bare different. Thus
a^3 b^2 and a^2 b^3 are different.
On the other hand, a^3 b^2 and b^2 a^3 are the same, since the powers of a and b in these
two terms are the same. The order of factors does not matter.
Thus,a^3 b^2 = a^3 × b^2 = b^2 × a^3 = b^2 a^3. Similarly, a^2 b^3 and b^3 a^2 are the same.
EXAMPLE 5 Express the following numbers as a product of powers of prime factors:
(i) 7 2 (ii) 4 3 2 (iii) 1000 (iv) 16000
SOLUTION
(i) 72 = 2 × 36 = 2 × 2 × 18
= 2 × 2 × 2 × 9
= 2 × 2 × 2 × 3 × 3 = 2^3 × 3^2
Thus, 72 = 2^3 × 3^2 (required prime factor product form)
TRY THESE
Express:
(i) 729 as a power of 3
(ii) 128 as a power of 2
(iii) 343 as a power of 7272
236
218
39
3