Question

The value of (994)1/3 according to binomial theorem is

Answer 1

Answer:

$(994)^{ \frac{1}{3} } \\ \\ (1000 - 6)^{ \frac{1}{3} } \\ \\ \bigg \{1000 \bigg( 1 - \frac{6}{1000} \bigg ) \bigg \}^{ \frac{1}{3} } \\ \\ \bigg \{10 ( 1 - 0.006 ) \bigg \}^{ \frac{1}{3} } \\ \\10 \bigg ( 1 - \frac{1}{3} \times 0.006 \bigg) \\ \\ 10(1 - 0.002) \\ \\ 10(0.998) \\ \\ \bf 9.98$

Answer 2

Given:

(994)^1/3

To Find:

Find the value using the binomial theorem

Solution:

When the power of the algebraic expression increases it becomes lengthy to find the value like (x+3)^66 is very hard to calculate that why we use binomial theorem for the expansion of term with greater powers or fractions, a binomial expansion should only contain two not similar terms if the power of the expansion is n then the total number of terms in the expansion will be (n+1).

The formula for expansion if the power is in the fraction is,

If n is negative or fraction also |x|<1, then the expansion will be,

$(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2 +\frac{n(n-1)(n-2)}{3!}x^3 +...$

Now to find the value of (994)^1/3, we will go as

$=(994)^{1/3}\\=(993+1)^{1/3}\\=1+\frac{993}{3}+\frac{\frac{1}{3} *\frac{-2}{3} }{2!}*993*993+\frac{\frac{1}{3}*\frac{-2}{3} *\frac{-5}{3} }{3!}*993*993*993+...\\=1+331-54780.5+60441151.67-...\\=9.97$

Hence, the value of (994)^1/3 is 9.97.