# Solve the given problem that involves binomial theorem

• chwala
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chwala
Gold Member
Homework Statement
See attached
Relevant Equations
Binomial theorem

part (a)

##(4+3x)^{1.5} = 2^3+ 9x+ \left[\dfrac {1}{2} ⋅ \dfrac {3}{2} ⋅\dfrac {1}{2}⋅\dfrac {1}{2}⋅9x^2\right]+ ...##

##(4+3x)^{1.5}=8+9x+\dfrac {27}{16} x^2+...##part (b)

##x≠-\dfrac {4}{3}##part (c)

##(8+9x+\dfrac {27}{16} x^2+...)(1+ax)^2 = \dfrac{107}{16} x^2##

...

##8a^2+18a+\dfrac {27}{16}=\dfrac{107}{16}##

##8a^2+18a=\dfrac{80}{16}##

##128a^2+288a-80=0##

##8a^2+18a-5=0##

##a_1=0.25##

##a_2= -2.5##

Bingo!

Any other way welcome guys...

The binomial expansion of $(1 + x)^\alpha$ is only valid for $|x| < 1$. $$(4 + 3x)^{1.5} = 4^{1.5}\left(1 + \frac{3x}{4}\right)^{1.5}.$$

chwala
pasmith said:
The binomial expansion of $(1 + x)^\alpha$ is only valid for $|x| < 1$. $$(4 + 3x)^{1.5} = 4^{1.5}\left(1 + \frac{3x}{4}\right)^{1.5}.$$
I should have expressed my answer as,

##|x|<\dfrac{4}{3}##

## What is the binomial theorem?

The binomial theorem describes the algebraic expansion of powers of a binomial. It states that (a + b)^n can be expanded into a sum involving terms of the form C(n, k) * a^(n-k) * b^k, where C(n, k) is a binomial coefficient.

## How do you calculate binomial coefficients?

Binomial coefficients, denoted as C(n, k) or "n choose k", are calculated using the formula C(n, k) = n! / [k! * (n - k)!], where "!" denotes factorial, the product of all positive integers up to that number.

## What are some common applications of the binomial theorem?

The binomial theorem is used in probability theory, combinatorics, and algebra. It helps in expanding expressions raised to a power, calculating probabilities in binomial distributions, and simplifying polynomial expressions.

## How do you expand (a + b)^n using the binomial theorem?

To expand (a + b)^n, write it as a sum of binomial terms: (a + b)^n = Σ [C(n, k) * a^(n-k) * b^k] for k = 0 to n. Each term involves a binomial coefficient, a power of 'a', and a power of 'b'.

## What is the significance of Pascal's Triangle in the binomial theorem?

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two directly above it. The rows of Pascal's Triangle correspond to the coefficients in the binomial expansion of (a + b)^n, making it a useful tool for quickly finding binomial coefficients.

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