Question

(32.1)¹/⁵,Find approximate value.

Answer 1

we have to find approximate value of (32.1)^1/5
using binomial expansion,
if y = (a ±b)ⁿ in such a way that y = aⁿ(1 + b/a)ⁿ
and 1 >>> b/a then, y ≈ aⁿ (1 + nb/a)

now, (32.1)^1/5

= (32 + 0.1)^1/5

= (32)^1/5{1 + 0.1/32}^1/5

= (2^5)^1/5 {1 + 1/320}^1/5

= 2(1 + 1/320)^1/5

here it is clearly shown that 1 >> 1/320 so, we can apply binomial expansion.

= 2(1 + 1/5 × 1/320)

= 2(1 + 1/1600)

= 2(1 + 0.000625)

= 2(1.000625)

= 2.001250

hence , (32.1)^1/5 ≈ 2.00125

Answer 2

Dear Student:

For finding we will define x =32

and Δx=0.1

and use,Δy=dy/dx*(Δx)

And,also Δy=f(x+Δx0-f(x)

See the attachment: