Write the powers to which the bases to be raised in the following. i

1.	Write the powers to which the bases to be raised in the following. i) 64 = 2xii) 100 = 5baiii) \(\frac{1}{81}=3^c\)iv) 100 = 102v) \(\frac{1}{256}\) = 4a
Answer» i) 64 = 2^x We know that 64 = 2 × 32 = 2 × 2 × 16 = 2 × 2 × 2 × 8 = 2 × 2 × 2 × 2 × 4 = 2 × 2 × 2 × 2 × 2 × 2 64 = 2⁶ ⇒ x = 6 ii) 100 = 5^b Here also 100 cannot be written as any power of 5. i.e., there exists no integer for b such that 5^b = 100 iii) \(\frac{1}{81}=3^c\) We know that 81 = 3 x 27 = 3 × 3 × 9 = 3 × 3 × 3 × 3 = 34 ∴ \(\frac{1}{81}=3^{-4}\) [∵a^-3 = \(\frac{1}{a^m}\)] ∴ c = – 4 iv) 100 = 10² z = 2 v) \(\frac{1}{256}\)= 4^a We know that 256 = 4 × 64 = 4 × 4 × 16 = 4 × 4 × 4 × 4 ∴ \(\frac{1}{256}\) = 4 ∴ a = – 4

Write the powers to which the bases to be raised in the following. i) 64 = 2xii) 100 = 5baiii) \(\frac{1}{81}=3^c\)iv) 100 = 102v) \(\frac{1}{256}\) = 4a

Answer»

i) 64 = 2^x

We know that

64 = 2 × 32

= 2 × 2 × 16

= 2 × 2 × 2 × 8

= 2 × 2 × 2 × 2 × 4

= 2 × 2 × 2 × 2 × 2 × 2

64 = 2⁶

⇒ x = 6

ii) 100 = 5^b

Here also 100 cannot be written as any power of 5.

i.e., there exists no integer for b such that 5^b = 100

iii) \(\frac{1}{81}=3^c\)

We know that 81 = 3 x 27

= 3 × 3 × 9

= 3 × 3 × 3 × 3

= 34

∴ \(\frac{1}{81}=3^{-4}\) [∵a^-3 = \(\frac{1}{a^m}\)]

∴ c = – 4

iv) 100 = 10²

z = 2

v) \(\frac{1}{256}\)= 4^a

We know that 256 = 4 × 64

= 4 × 4 × 16

= 4 × 4 × 4 × 4

∴ \(\frac{1}{256}\) = 4

∴ a = – 4