Problem 65 - Intermediate Algebra

Question

Evaluate each expression. See Example \(8 .\) $$ 4 \cdot 2^{3} $$

Step-by-Step Solution

Answer

The expression evaluates to 32.

Step 1: Identify the Base and Exponent

The expression given is \( 4 \cdot 2^{3} \). Here, 2 is the base and 3 is the exponent in the term \( 2^3 \).

Step 2: Compute the Exponential Value

Calculate \( 2^3 \), which means multiplying 2 by itself 3 times: \( 2 \times 2 \times 2 = 8 \).

Step 3: Multiply by the Scalar

Now take the result from the exponential calculation, which is 8, and multiply it by the factor outside the exponent, 4: \( 4 \times 8 \).

Step 4: Perform the Multiplication

Multiply 4 by 8 to get the final result: \( 4 \times 8 = 32 \).

Key Concepts

Algebra Basics

Algebra is the branch of mathematics dealing with symbols and rules for manipulating those symbols. These symbols represent numbers in formulas and equations, allowing us to solve for unknowns. In our exercise, the expression \( 4 \cdot 2^{3} \) involves understanding how to manipulate these numbers and symbols according to algebraic rules. Algebra often requires breaking down expressions into understandable components, such as:

• Identifying constants: In \( 4 \cdot 2^{3} \), 4 and 2 are constants.
• Recognizing operations: "\( \cdot \)" denotes multiplication here.
• Understanding expressions: \( 2^{3} \) is an exponential operation.

Algebra strengthens our ability to think logically and solve complex problems by understanding how numbers interact. This is vital for efficiently tackling both simple and advanced mathematical problems.

Order of Operations

The order of operations is crucial for solving mathematical expressions correctly. It helps us determine which operations to perform first to ensure accuracy. In mathematics, we often use the acronym PEMDAS to remember the sequence:

• Parentheses: Perform calculations inside parentheses first.
• Exponents: Solve exponents or powers next, such as \( 2^3 \).
• Multiplication and Division: Proceed with multiplication and division, moving from left to right.
• Addition and Subtraction: Finally, handle addition and subtraction, moving from left to right as well.

In our example, \( 4 \cdot 2^{3} \), the correct order involves evaluating the exponent \( 2^{3} \) first. Once simplified to 8, you then multiply by 4 to deliver the final value. This careful sequencing prevents mistakes and ensures your answer is accurate.

Understanding Multiplication

Multiplication is one of the fundamental arithmetic operations and represents repeated addition. For instance, \( 4 \times 8 \) is essentially adding 8 to itself 4 times: \( 8 + 8 + 8 + 8 = 32 \). It is streamlined and efficient, allowing us to quickly calculate results without cumbersome repetitive addition. In multiplication:

• Factors are the numbers being multiplied together. In \( 4 \cdot 8 \), 4 and 8 are factors.
• The result of multiplication is called the product. Here, 32 is the product.
• Understanding basic multiplication is critical for higher mathematical concepts, like algebra and calculus.

The process of multiplying two numbers involves growing one number into bigger equal parts guided by another. Grasping multiplication is essential not only for elementary arithmetic but also for interpreting and solving more intricate mathematical expressions and problems.