Problem 132 - Algebra A Combined Function

Question

Simplify each expression. Assume that variables represent positive integers. $$ a^{-4 m} \cdot a^{5 m} $$

Step-by-Step Solution

Answer

The simplified expression is \( a^m \).

Step 1: Identify the Base and Exponents

The expression is given as \( a^{-4m} \cdot a^{5m} \). Here, the base is \( a \), and the exponents are \(-4m\) and \(5m\).

Step 2: Apply the Product of Powers Property

Using the product of powers property, which states that \( a^m \cdot a^n = a^{m+n} \), add the exponents: \(-4m + 5m\).

Step 3: Simplify the Exponents

Simplify the sum of the exponents: \(-4m + 5m = m\).

Step 4: Write the Simplified Expression

Replace the original exponents with the simplified exponent to get the final expression: \( a^{m} \).

Key Concepts

Product of Powers Property

The "product of powers property" is a fundamental rule in mathematics that helps simplify expressions containing exponents. When multiplying two expressions with the same base, you can combine them by adding their exponents. This property is expressed in the formula:

• \( a^m \cdot a^n = a^{m+n} \)

This means if you have the same base, \( a \), raised to two different powers, \( m \) and \( n \), you simply add those exponents together to simplify. For instance, if you come across the expression \( a^3 \cdot a^5 \), applying this property would result in \( a^{3+5} = a^8 \).
This property is particularly useful in equations and algebraic manipulations involving powers, allowing you to simplify and solve them more easily. Remember, this property only applies when the bases are the same.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This often means combining like terms or applying mathematical properties to make the expression easier to read or work with. In our example with \( a^{-4m} \cdot a^{5m} \),
we have already identified the use of the product of powers property to add the exponents.Here’s how you simplify:

• First, align all terms sharing the same base.
• Apply relevant property or rule, such as adding exponents with the same base.
• Calculate and write out the combined or reduced result.

In this case, after applying the product of powers property, you sum the exponents: \( -4m + 5m = m \). This results in the simplified expression \( a^m \).
By simplifying expressions, you make problems easier to manage and understand, whether for further calculations or for finding a solution.

Properties of Exponents

Exponents have several important properties that simplify calculations and algebraic manipulations. Here are some crucial properties:

• Product of Powers: As discussed, multiply with the same base by adding exponents: \( a^m \cdot a^n = a^{m+n} \).
• Power of a Power: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
• Quotient of Powers: When dividing with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Negative Exponent: A negative exponent indicates a reciprocal: \( a^{-n} = \frac{1}{a^n} \).
• Zero Exponent: Any non-zero base raised to the power of zero is 1: \( a^0 = 1 \).

These properties are foundational tools for anyone studying algebra and are regularly applied in simplifying expressions and solving equations. Understanding these properties will enhance your ability to work efficiently with exponents across different mathematical contexts.