Problem 70 - Algebra A Combined Function

Question

Evaluate each expression. $$ 4^{3} $$

Step-by-Step Solution

Answer

The expression \(4^3\) evaluates to 64.

Step 1: Understand the Expression

We need to evaluate the expression \(4^3\). This implies calculating the power, where 4 is the base, and 3 is the exponent.

Step 2: Calculate the Power

The expression \(4^3\) means we are multiplying the base 4 by itself for a total of three factors. Therefore, it can be expanded to \(4 \times 4 \times 4\).

Step 3: Multiply the Factors

First, multiply the first two numbers: \(4 \times 4 = 16\). Then multiply the result with the third number: \(16 \times 4 = 64\).

Step 4: State the Final Result

Thus, the evaluation of \(4^3\) results in 64.

Key Concepts

Power and Exponentiation

When dealing with powers, the term "exponentiation" refers to the mathematical operation involving powers. In simple terms, it is a shorthand method to represent repeated multiplication of the same number.
For example, instead of saying "multiply 4 by itself 3 times," we say "4 raised to the power of 3" or just "4 cubed."
The expression is written as \(4^3\), where the number at the bottom (4) is the base, and the small number above it (3) is the exponent.

• Exponents indicate how many times the base is used as a factor.
• Exponentiation simplifies expressions and calculations.

Understanding exponentiation is vital in math because it compactly expresses numbers and facilitates calculations, especially in algebra and beyond.

Base and Exponent

In an expression like \(4^3\), the base and the exponent are two key components. The base is the larger number written at the bottom (4 in this case), and it tells you what number is being multiplied.
Meanwhile, the exponent is the smaller number written at the top right corner (3 here) and indicates the number of times the base is multiplied by itself.
The base can be any number, and the exponent is typically a positive integer, though it can be other values too.

• The base represents the repeated factor.
• The exponent shows how frequently the base is multiplied.

For instance, in \(4^3\), the calculation involves three 4's being multiplied. Thus, learning about the base and exponent helps students understand not just multiplication but how numbers grow exponentially.

Multiplication Process

When calculating powers, like \(4^3\), the multiplication process comes into play.
To evaluate \(4^3\), you begin by expanding the expression into a multiplication sentence: \(4 \times 4 \times 4\).
Start with the first two numbers: \(4 \times 4\) gives 16. Then, multiply 16 by the last 4: \(16 \times 4\) results in 64.

• The process demonstrates step-by-step multiplication.
• Sequential computing helps avoid mistakes.

Breaking it down into steps not only solves the problem efficiently but also instills a solid understanding of the multiplication involved in exponentiation. With practice, these skills enhance a student's problem-solving abilities in higher mathematics.