Question

(80)¹/⁴,Find approximate value.

Answer 1

we have to find out approximate value of (80)¼
using binomial expansion , if y = (a + b)ⁿ is a expression in such a way that y = aⁿ (1 + b/a)ⁿ
where 1 >> b/a then, y ≈ aⁿ (1 + nb/a)

here, (80)¼

= (81 - 1)¼

= {81(1 - 1/81)}¼

= (81)¼ (1 - 1/81)¼

here it is clear that 1 >> 1/81
so, we can definitely apply binomial expansion.

= (3⁴)¼ (1 - 1/4 × 1/81)

= 3(1 - 1/324)

= 3(1 - 0.003086)

= 3(0.996914)

= 2.990742

hence, approximate value of (80)¼ ≈ 2.990742

Answer 2

Dear Student:

For finding we will define x =81

and Δx=-1

and use,Δy=dy/dx*(Δx)

And,also Δy=f(x+Δx)-f(x)

See the attachment: