Problem 75 - Algebra A Combined Function

Question

Simplify each expression. $$ b^{4} b^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \( b^6 \).

Step 1: Identify the Base

The expression given is \( b^4 b^2 \). Notice that both terms have the same base, which is \( b \).

Step 2: Identify the Exponents

In the expression \( b^4 b^2 \), the exponents are 4 and 2 respectively. These are the powers to which the base \( b \) is raised.

Step 3: Apply the Rule for Multiplying Powers

When multiplying powers with the same base, you add the exponents. Therefore, \( b^4 b^2 = b^{4+2} \).

Step 4: Calculate the Resulting Exponent

After applying the rule for exponents, we add the exponents: \( 4 + 2 = 6 \). Hence, the expression is \( b^6 \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial step in algebra that makes them easier to understand and work with. When you simplify an expression, you aim to reduce it to its simplest form by applying mathematical operations and the rules of algebra.

Here’s how you can think about it when dealing with powers and exponents:

• Look for common terms: Identify components that can be combined and apply the same rules to them.
• Apply arithmetic operations: Use operations like addition or multiplication to combine terms.
• Use exponent rules: Recognize patterns in the expression where exponent rules, such as multiplication, can simplify the expression.

In the example provided, the expression is simplified by combining terms with the same base, making it easier to evaluate or apply further operations.

Multiplying Powers

Multiplying powers is a fundamental concept in algebra which involves dealing with expressions where the same base is raised to different exponents. When you multiply powers that have the same base, you can simplify the expression by adding the exponents together.

Consider the rule for multiplying powers:

• If the base is the same, just add the exponents: If you have an expression like \( b^m \times b^n \), the result will be \( b^{m+n} \).

In our exercise, where the expression is \( b^4 \times b^2 \), you apply this rule:
- The base is the same, enabling the use of the rule for multiplying powers.
- You add the exponents: 4 (from \( b^4 \)) and 2 (from \( b^2 \)), resulting in \( b^{4+2} \), or simply \( b^6 \). This reduces the expression to a single power of the base.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and arithmetic operations such as addition, subtraction, multiplication, and exponentiation. Understanding how to work with them is crucial for solving many problems in mathematics.

An algebraic expression may include several terms that can be combined or simplified using rules for arithmetic and exponents. Here are key elements to focus on:

• Variables: Symbols such as \( b \) representing numbers in an expression.
• Operations: Operations include addition, subtraction, multiplication, and division, plus exponentiation, which is particularly important when dealing with powers.
• Combining like terms: Simplify expressions by grouping and combining terms that have the same variable and exponents.

In the example provided, the algebraic expression \( b^4 \times b^2 \) is simplified using these principles, focusing on the operation of multiplication and the rule of adding exponents to produce a simpler expression: \( b^6 \). Understanding these elements helps demystify algebraic expressions and aids in solving more complex equations.