Exponent Power of a Power Rule

The power of a power rule is one of the key rules for working with exponents. It states that when you raise an exponential expression to another power, you multiply the exponents. In mathematical terms, for any real number \( a \) and any exponents \( m \) and \( n \):

Why Does This Rule Work?

Let’s break it down step by step:

1. \[ a^m = \underbrace{a \times a \times \cdots \times a}_{m\text{ times}} \]
2. \[ \bigl(a^m\bigr)^n = \underbrace{a^m \times a^m \times \cdots \times a^m}_{n\text{ times}} \]
3. \[ a^m \times a^m \times \cdots \times a^m = a^{m + m + \cdots + m} = a^{m \times n} \]

Examples

1. Consider \(\bigl(2^3\bigr)^4\):

   • First, recognize that \(2^3 = 2 \times 2 \times 2\).
   • Then, raising \(2^3\) to the 4th power means: \[ \bigl(2^3\bigr)^4 = 2^{3 \times 4} = 2^{12} \]
   • Calculating \(2^{12}\) gives: \[ 2^{12} = 4096 \]
2. \[ \bigl(x^2\bigr)^5 = x^{2 \times 5} = x^{10} \]

When to Use This Rule

• Simplifying Expressions: Whenever you have a power raised to another power, you can simplify the expression quickly by multiplying the exponents.
• Solving Equations: It helps in solving equations that involve exponential expressions by reducing the complexity of the exponents.
• Algebraic Manipulations: It is frequently used in algebra to rewrite expressions in a more manageable form.

Important Note

While the rule \(\bigl(a^m\bigr)^n = a^{m \times n}\) holds for many cases, be cautious when dealing with:

• Negative bases: Especially when the exponents are not whole numbers.
• Fractional or irrational exponents: Ensure the base is within the domain where the expression is defined.

Understanding this rule makes it much easier to work with exponential expressions in algebra and calculus.

Latex

\bigl(a^m\bigr)^n = a^{m \times n}