Question

Approximate value of (31)¹/⁵ is.......,Select correct option from the given options.
(a) 2.01
(b) 2.1
(c) 2.0125
(d) 1.9875

Answer 1

we have to find out approximate value of $\bf{(31)^{1/5}}$

using binomial expansion ,
if y = (1 ± a)ⁿ is given and 1 >> a then, y ≈ 1 ± na


here, $\bf{(31)^{1/5}}$

$\implies\bf{(32-1)^{1/5}}$

$\implies\bf{(32)^{1/5}(1 - 1/32)^{1/5}}$

$\implies\bf{2(1-1/32)^{1/5}}$

here, we see 1 >> 1/32

$\implies\bf{2(1-1/5\times1/32)}$

$\implies\bf{2(1-1/160)$

$\implies\bf{2(1-0.00625)}$

$\implies\bf{2\times0.99375}$

$\implies\bf{1.9875}$

hence, option (d) is correct

Answer 2

Hello,

Answer: Option D ( 1.9875)

Solution:

let x = (31)¹/⁵

$y = {(x)}^{ \frac{1}{5} } \\ \\ y - dy \: = {(x - dx)}^{ \frac{1}{5} } \\ \\ y - dy = {(32 - 1)}^{ \frac{1}{5} }$
here dy and dx are slightly change in both values

Now y = (32)¹/⁵ = 2

dy = 1

dy = f'(x) dx

f'(x) =
$\frac{1}{5 \: {x}^{ \frac{4}{5} } } \times dx \\ \\ = \frac{1}{5 \: ({32})^{ \frac{4}{5} } } \times 1\\ \\ = \frac{1}{5 \times 16} \\ = \frac{1}{80} \\ \\ dy = 0.0125$
Approximate value of (31)¹/⁵ = y-dy

= $2 - 0.0125 \\ = 1.9875$
So, option d is correct answer.

Hope it helps you