Question

.0025 =2/5^a × 10^b​

Answer 1

Given :    0.0025 = [2/5]^a ×10^b​

To Find : Value of a and b

Solution:

0.0025 = [2/5]ᵃ ×10ᵇ

=> 0.0025 =(2ᵃ/5ᵃ) × 10ᵇ

=> 0.0025 =(2ᵃ/5ᵃ) ×(2 × 5 )ᵇ

=>  0.0025 =(2ᵃ/5ᵃ) × 2ᵇ × 5ᵇ

=>  0.0025 = 2ᵃ⁺ᵇ ₓ 5ᵇ⁻ᵃ

=>   25/10000  = 2ᵃ⁺ᵇ ₓ 5ᵇ⁻ᵃ

=> 1/400  =   2ᵃ⁺ᵇ ₓ 5ᵇ⁻ᵃ

=>   1/(2* 2 * 2 * 2 * 5 * 5)  =  2ᵃ⁺ᵇ ₓ 5ᵇ⁻ᵃ

=> 1/(2⁴ * 5²) =  2ᵃ⁺ᵇ ₓ 5ᵇ⁻ᵃ

=> 2⁻⁴ * 5⁻² = 2ᵃ⁺ᵇ ₓ 5ᵇ⁻ᵃ

2 and 5 are co prime

Hence equating powers of corresponding bases

=> a + b = - 4      

   b - a  =  - 2

=> 2b = - 6

=> b = - 3

 => a = - 1

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