Problem 58 - Elementary Algebra

Question

\( 3^{3}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

The problem requires finding the value of the exponential expression \( 3^{3} \). This means multiplying the base, 3, by itself a total of 3 times.

Step 2: Set Up the Expression

Write the expression in its expanded form: \( 3^{3} = 3 \times 3 \times 3 \).

Step 3: Multiply the Terms

First, multiply the first two terms: \( 3 \times 3 = 9 \). Then multiply the result by the remaining 3: \( 9 \times 3 = 27 \).

Step 4: Write the Final Answer

The value of the expression \( 3^{3} \) is 27.

Key Concepts

Exponents

Exponents are a shorthand way of expressing repeated multiplication of the same number. For example, in the expression \(3^3\), the number 3 is the base and the exponent is 3. This means you multiply the base, 3, by itself three times:
\(3 \times 3 \times 3\).

The exponent tells you how many times to use the base in the multiplication process.
Here are some important points to remember about exponents:

• The expression \(a^n\) means 'a' is multiplied by itself 'n' times.
• Any number raised to the power of 1 is the number itself, e.g., \(5^1 = 5\).
• Any number raised to the power of 0 is 1, e.g., \(7^0 = 1\). This applies to any non-zero number.

Developing a good understanding of exponents can simplify many aspects of higher-level math.

Multiplication

Multiplication is one of the four basic mathematical operations. It involves combining equal groups.
For instance, in the example of \(3^3\), you are combining three groups of 3. Here’s how you can visualize this:

• The first step is to combine the first two 3's:
  
  \(3 \times 3 = 9\).
• Next, you take the result (9) and multiply it by the remaining 3:
  
  \(9 \times 3 = 27\).

The grand total is 27. That’s how multiplication helps in calculating powers or exponents.
Multiplying terms one step at a time will make it easier to solve even more complicated expressions later.

Mathematical Operations

Mathematical operations include addition, subtraction, multiplication, and division.
Each operation has its rules and uses. When dealing with exponents, multiplication is usually the primary operation.

Here's how combining these concepts can help in solving problems:

• First, translate the exponent into a multiplication problem. For example, \(3^3\) translates to \(3 \times 3 \times 3\).
• Multiply the terms step-by-step to get your result. Simplifying problems into smaller steps, as shown, makes complex calculations more manageable.

Relying on understanding simple components of mathematical operations can vastly improve accuracy and efficiency while solving problems.

Remember: getting comfortable with these basic operations is crucial for tackling advanced math topics successfully.