Question

Simplify first three terms in the given expansion:
$\rm (5-3x)^{-1/3}$

Answer 1

Answer:

First 3 terms of the given expansion are  5⁻¹/₃{ 1 + x/5 + 18x²/25 + .....)

Step-by-step explanation:

Hi,

Given (5 - 3x)⁻¹/₃

= 5⁻¹/₃( 1 - 3x/5)⁻¹/₃

The above expansion ( 1 - 3x/5)⁻¹/₃ is very much similar to binomial

expansion (1 + y)ⁿ

here y = -3x/5 and n = -1/3

The first 3 terms of the  expansion ( 1 - 3x/5)⁻¹/₃ will be

1 + ⁿC₁y + ⁿC₂y² + ....

So first term of the expansion is 1

Second term of the expansion is ⁿC₁y = ny

= (-1/3)(-3x/5) = x/5

Third term in the expansion is ⁿC₂y²

= n(n-1)y²/2

= (-1/3)(-1/3-1)(-3x/5)²/2

= 1/3*4/3*9x²/50

=18x²/25

So, the first 3 term of the expression (5 - 3x)⁻¹/₃ will

= 5⁻¹/₃( 1 - 3x/5)⁻¹/₃

= 5⁻¹/₃{ 1 + x/5 + 18x²/25 + .....)

Hope, it helps !