Question

Find the approximate value of (1.999)⁵.

Answer 1

(1.999)⁵ = (2 – 0.001)⁵

Let x = 2 and Δx = – 0.001

Let y = x⁵

Differentiating both sides w.r.t, x, we get

`"dy"/"dx"` = 5x⁴ = 5(2) = 80

Now Δy = `("dy"/"dx") * Δx`

= 80 · (– 0.001)

= – 0.080

∴ (1.999)⁵ = y + Δy

= x⁵ – 0.080

= (2)⁵ – 0.080

= 32 – 0.080

= 31.92

Hence, approximate value of (1.999)⁵ is 31.92.