1. What is Binomial Expansion?

The binomial theorem describes the algebraic expansion of powers of a binomial (a + b). It provides a way to expand expressions of the form (a + b)ⁿ without having to perform all the multiplication.

2. How Does the Calculator Work?

The calculator uses the binomial expansion formula:

Where:

• \( a \) — First term in the binomial
• \( b \) — Second term in the binomial
• \( n \) — Power of the expansion
• \( k \) — Term number (0 to n)
• \( \binom{n}{k} \) — Binomial coefficient (n choose k)

Explanation: The calculator computes the specific term in the expansion using the binomial coefficient multiplied by the appropriate powers of a and b.

3. Importance of Binomial Expansion

Details: Binomial expansion is fundamental in algebra, probability, and calculus. It's used in probability theory (binomial distribution), series approximations, and polynomial expansions.

4. Using the Calculator

Tips: Enter values for a and b (can be any real numbers), the power n (must be non-negative integer), and the term number k (must be between 0 and n).

5. Frequently Asked Questions (FAQ)

Q1: What is the binomial coefficient?
A: The binomial coefficient C(n,k) represents the number of ways to choose k elements from a set of n elements.

Q2: What if k is greater than n?
A: The binomial coefficient is zero when k > n, so the term value would be zero.

Q3: Can I use this for negative exponents?
A: No, this calculator only works for non-negative integer exponents (n ≥ 0).

Q4: What about fractional exponents?
A: For fractional exponents, you would need an infinite series expansion (Taylor series), not covered by this calculator.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact for the given inputs, though displayed with 4 decimal places for readability.