Exponent Power of Product Rule

The Power of a Product Rule for exponents states that when you raise a product to an exponent, you can distribute the exponent to each factor in the product. In mathematical terms, for any numbers (or expressions) \(a\) and \(b\) and any exponent \(n\), the rule is written as:

How It Works

1. \[ (ab)^n = \underbrace{(ab) \cdot (ab) \cdot \ldots \cdot (ab)}_{n \text{ times}} \]
2. \[ (ab)^n = \underbrace{a \cdot a \cdot \ldots \cdot a}_{n \text{ times}} \cdot \underbrace{b \cdot b \cdot \ldots \cdot b}_{n \text{ times}} = a^n \cdot b^n \]

Examples

• \[ (2 \cdot 3)^4 = 6^4 = 6 \times 6 \times 6 \times 6 = 1296 \]\[ 2^4 \cdot 3^4 = 16 \cdot 81 = 1296 \]
• \[ (xy)^3 = x^3 \cdot y^3 \]
• \[ (ab)^{-n} = a^{-n} \cdot b^{-n} \]\[ (ab)^{\frac{1}{2}} = \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \]

Important Notes

• \[ (a + b)^n \neq a^n + b^n \]

  (There are special cases, but the rule does not hold in general.)

• Why It’s Useful:
  This rule simplifies expressions and is used extensively in algebra for solving equations, expanding expressions, and simplifying terms involving exponents.

Summary

The Power of a Product Rule is a straightforward yet powerful tool in algebra that allows you to simplify expressions by distributing an exponent over a product:

By understanding and applying this rule, you can simplify many algebraic expressions and solve problems involving exponents more efficiently.

Latex

(ab)^n = a^n \cdot b^n