Problem 130 - Algebra A Combined Function

Question

Simplify each expression. Assume that variables represent positive integers. $$ b^{9 a} b^{4 a} $$

Step-by-Step Solution

Answer

The expression simplifies to \(b^{13a}\).

Step 1: Identify the Expression

The given expression to simplify is \(b^{9a} b^{4a}\). This expression involves multiplying two powers with the same base \(b\).

Step 2: Apply the Laws of Exponents

When multiplying powers with the same base, we add the exponents. According to the law of exponents, \(b^m \times b^n = b^{m+n}\).

Step 3: Simplify the Expression

Add the exponents \(9a\) and \(4a\) together: \[ 9a + 4a = 13a \]So, the expression simplifies to \(b^{13a}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is about making mathematical expressions easier or more compact without changing their value. This means writing them in a form that is easier to understand or work with while maintaining the same mathematical meaning. When you simplify an expression, you often look for ways to combine like terms or use mathematical rules to reduce complexity.

• First, identify the parts of the expression you can simplify. In algebra, this includes terms that can be combined or exponents that can be added.
• Next, apply any relevant mathematical operations or rules, such as addition or multiplication of terms.

For example, in the expression \(b^{9a} b^{4a}\), our task is to simplify it by combining the powers of \(b\) using exponent rules. The simplification reduces the expression to the simplest equivalent form, \(b^{13a}\). This keeps the expression easier to work with and understand in future calculations.

Laws of Exponents

The laws of exponents are rules that describe how to handle exponents in mathematical expressions. These laws make it easier to work with powers, especially when simplifying expressions or solving equations.Some key laws of exponents include:

• Product of Powers Rule: This rule states that when you multiply two powers with the same base, you add the exponents: \(b^m \times b^n = b^{m+n}\).
• Quotient of Powers Rule: This rule is used when you're dividing two powers with the same base: \(b^m / b^n = b^{m-n}\).
• Power of a Power Rule: This rule applies when raising a power to another power, and you multiply the exponents: \((b^m)^n = b^{m \times n}\).

These laws provide a framework for simplifying expressions and ensuring that calculations are accurate and reliable. In the given problem, applying the Product of Powers Rule allowed us to simplify \(b^{9a} b^{4a}\) to \(b^{13a}\). Understanding these core exponent rules is crucial for efficient problem-solving in algebra.

Multiplying Powers with Same Base

Multiplying powers with the same base involves using exponent rules to combine them into a simpler form. This process uses one of the fundamental laws of exponents, specifically the Product of Powers Rule. When you encounter two or more exponent terms with the same base, this rule helps simplify the expression by adding the exponents.

• For instance, with the expression \(b^{9a} \times b^{4a}\), we identify the base as \(b\).
• Using the exponent rule, we simply add the exponents: \(9a + 4a\).
• This gives us \(b^{13a}\).

The benefit of this is a more streamlined expression that's easier to manage in calculations. By transforming \(b^{9a} \times b^{4a}\) into \(b^{13a}\), we've condensed the expression, making future operations, like evaluation or further algebraic manipulations, much simpler. Mastering this concept is essential for handling more complex algebraic problems efficiently.