How to Expand a Bracket with Power?

Expanding a bracket with a power involves raising the entire expression inside the bracket to that power. According to the reference, when you have terms in the bracket raised to a power, you must highlight the terms in the bracket and the power, then click expand. This often involves using the binomial theorem or, for simpler powers, repeated multiplication. Let's explore how to do this with examples.

Understanding the Concept

When we say "expand (a + b)^n", we mean to write it out without the power, as a sum of terms. This process becomes more involved as n increases.

Methods for Expansion

There are two primary methods for expanding a bracket raised to a power:

1. Repeated Multiplication: This method is straightforward for smaller powers (like squared or cubed).
2. Binomial Theorem: This provides a general formula for any power.

1. Repeated Multiplication

For example, let's expand (x + 2)^2. This means (x + 2)(x + 2).

• First, multiply each term in the first bracket by each term in the second bracket:
  • x * x = x²
  • x * 2 = 2x
  • 2 * x = 2x
  • 2 * 2 = 4
• Then, combine like terms:
  • x² + 2x + 2x + 4 = x² + 4x + 4

Therefore, (x + 2)^2 = x^2 + 4x + 4

2. Binomial Theorem

The binomial theorem provides a formula for expanding (a + b)^n for any positive integer n:

(a + b)^n = ∑_(k=0)^n (n choose k) a^(n-k) b^k

Where (n choose k) is the binomial coefficient, calculated as n! / (k!(n-k)!)

Let's expand (x + y)^3 using the Binomial Theorem:

• n = 3, a = x, b = y

  (x + y)^3 = (3 choose 0)x^3y^0 + (3 choose 1)x^2y^1 + (3 choose 2)x^1y^2 + (3 choose 3)x^0y^3

• Calculate the binomial coefficients:

  • (3 choose 0) = 3! / (0!3!) = 1
  • (3 choose 1) = 3! / (1!2!) = 3
  • (3 choose 2) = 3! / (2!1!) = 3
  • (3 choose 3) = 3! / (3!0!) = 1

• Substitute the binomial coefficients:

  (x + y)^3 = 1*x^3*y^0 + 3*x^2*y^1 + 3*x^1*y^2 + 1*x^0*y^3

• Simplify:

  (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

Examples

Here are a few more examples to illustrate the process:

• (2x - 1)^2 = (2x - 1)(2x - 1) = 4x² - 4x + 1
• (a + b)^4 = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴ (Using the Binomial Theorem or Pascal's Triangle).
• (x - 3)^3 = x³ - 9x² + 27x - 27 (Using the Binomial Theorem or repeated multiplication).

Summary

Power	Expansion Technique	Example	Result
2	Repeated Multiplication	(x + 1)^2	x2 + 2x + 1
3	Binomial Theorem/Pascal's Triangle	(a - b)^3	a3 - 3a2b + 3ab2 - b3
4	Binomial Theorem/Pascal's Triangle	(2 + y)^4	16 + 32y + 24y2 + 8y3 + y4