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:

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

How It Works

1. Definition of Exponentiation:
   Raising a product (ab) to the power (n) means multiplying (ab) by itself (n) times:

   $$(ab)^n = \underbrace{(ab) \cdot (ab) \cdot \ldots \cdot (ab)}_{n \text{ times}}$$

2. Associative and Commutative Properties:
   Since multiplication is both associative (the grouping of factors does not affect the product) and commutative (the order of factors does not matter), you can rearrange the factors. This means you can group all the (a)‘s together and all the (b)‘s together:

   $$(ab)^n = \underbrace{a \cdot a \cdot \ldots \cdot a}_{n \text{ times}} \cdot \underbrace{b \cdot b \cdot \ldots \cdot b}_{n \text{ times}} = a^n \cdot b^n$$

Examples

• Numeric Example:
  Evaluate ((2 \cdot 3)^4).

  $$(2 \cdot 3)^4 = 6^4 = 6 \times 6 \times 6 \times 6 = 1296$$

  Using the rule:

  $$2^4 \cdot 3^4 = 16 \cdot 81 = 1296$$

• Algebraic Example:
  If (x) and (y) are variables, then:

  $$(xy)^3 = x^3 \cdot y^3$$

• With Negative or Fractional Exponents:
  For negative exponents:

  $$(ab)^{-n} = a^{-n} \cdot b^{-n}$$

  For fractional exponents (assuming (a) and (b) are nonnegative if the exponent represents a root):

  $$(ab)^{\frac{1}{2}} = \sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$$

Important Notes

• Not for Sums:
  This rule applies only to products. A common mistake is to try to apply it to sums, but in general:

  $$(a + b)^n \neq a^n + b^n$$

  (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:

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

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