Problem 84 - Prealgebra

Question

Perform the indicated operations. $$3^{3}$$

Step-by-Step Solution

Answer

The value of \(3^3\) is 27.

Step 1: Understand the Power

The expression \(3^3\) means 3 is raised to the power of 3, which can be interpreted as multiplying 3 by itself three times.

Step 2: Write Out the Multiplication

Write the expression \(3^3\) as a multiplication: \(3 \times 3 \times 3\).

Step 3: Perform the Multiplication

Multiply the numbers step-by-step: first \(3 \times 3 = 9\), then take the result and multiply by 3 again: \(9 \times 3 = 27\).

Step 4: State the Final Answer

The value of \(3^3\) is 27.

Key Concepts

Understanding the Power of a Number

To understand the power of a number, also known as an exponent, let's break it down. In mathematics, when you see an expression like \(3^3\), it signifies that the base number, in this case, 3, is multiplied by itself a certain number of times. The small number written above and to the right of the base, called the exponent, tells you how many times to multiply the base by itself. For \(3^3\), 3 is the base and the exponent is also 3, meaning you multiply 3 by itself two additional times.

• The base is the number being multiplied.
• The exponent indicates the number of times multiplication occurs.

So, when you see \(3^3\), think of it as 3 times 3 times 3. Understanding this will help simplify complex arithmetic operations involving powers.

The Role of Multiplication

Multiplication is at the heart of understanding and computing powers of numbers. Let's look at how multiplication works with powers. When you have \(3^3\), it's represented as \(3 \times 3 \times 3\). This process means, starting from the left, you take the base number and successively multiply it by itself.

• Start by multiplying the first two 3s: \(3 \times 3\).
• This gives you 9.
• Next, multiply this result by the third 3: \(9 \times 3\).
• The final result is 27.

This step-by-step multiplication helps in understanding and solving powers. By repeatedly applying multiplication, we build our understanding of exponents and improve calculation skills.

The Basics of Arithmetic Operations

Arithmetic operations are fundamental in math, and they include addition, subtraction, multiplication, and division. In the context of powers of numbers like \(3^3\), multiplication is the key operation. But beyond this, understanding all four operations allows for deeper comprehension and greater numerical fluency.

• **Addition and Subtraction:** Though not directly used in calculating \(3^3\), they are foundational for understanding relationships between numbers.
• **Multiplication:** As shown in the power of a number, this is repeatedly applied to solve expressions like \(3^3\).
• **Division:** It's often used to break down and simplify problems, although not directly needed for solving what's given here.

Recognizing how each operation interplays within mathematics builds a solid base for more advanced topics.