Question

binomial expansion of 10004 to the power 1/4

Answer 1

we have to find $\sqrt[4]{10004}$ with help of binomial expansion.

it is given that, $\sqrt[4]{10004}$

= $\left(10000+4\right)^{1/4}$

= $(10000)^{1/4}\left(1+\frac{4}{10000}\right)^{1/4}$

= $(10^4)^{1/4}(1+0.0004)^{1/4}$

= $10(1 + 0.0004)^{1/4}$

we know from binomial expansion,

(1 + x)ⁿ = 1 + nx + n(n -1)x²/2! + n(n - 1)(n - 2)x³/3! + .....

here, x = 0.0004 , n = 1/4

so, (1 + 0.0004)^{1/4} = 1 + (1/4)(0.0004) + (1/4)(1/4 -1 )(0.0004)²/2! + (1/4)(1/4-1)(1/4 - 2)(0.0004)³/3! + ........

= 1 + 0.0001 - 1.5 × 10^-8 + 3.5 × 10^-12 + .....

[ here 1.5 × 10^-8 and 3.5 × 10^-12 are very small value. on comparison 0.0001 , both are negligible . so, we assume in expansion after 0.0001, is zero. ]

≈ 1 + 0.0001 = 1.0001

so, 10(1 + 0.0004)^{1/4} ≈ 10(1.0001)

≈ 10.001