Problem 89 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$(0.5)^{2}$$

Step-by-Step Solution

Answer

The solution is 0.25.

Step 1: Identify the Base and Exponent

The expression given is \( (0.5)^{2} \), where 0.5 is the base and 2 is the exponent. This means you need to square 0.5.

Step 2: Apply Exponent Multiplication Rule

To square a number means to multiply it by itself. Therefore, compute \( 0.5 imes 0.5 \).

Step 3: Perform the Multiplication

Calculate \( 0.5 imes 0.5 = 0.25 \). The square of 0.5 is 0.25.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a certain power, known as the exponent. This operation is a fundamental building block in mathematics and is frequently used in various fields, including algebra, calculus, and physics.

• When you see an expression like \( (a)^{b} \), it means "a raised to the power of b".
• The base is the number being multiplied.
• The exponent signifies how many times the base is multiplied by itself.

Exponentiation is often represented as the repeated multiplication of the base. For example, if the base is 0.5 and the exponent is 2, you are essentially finding \( 0.5 \times 0.5 \).This operation simplifies calculations and helps compactly represent very large or very small numbers. Understanding exponentiation is crucial for solving equations and analyzing exponential growth or decay.

Base and exponent

In the expression \( (0.5)^{2} \), 0.5 is referred to as the base, and 2 is the exponent. These components are essential for understanding the operation being performed and interpreting what the expression means. The base is the number you start with, the primary subject of the multiplication process.

• For example, in \( (0.5)^{2} \), the base is 0.5.
• The base is the "foundation", determining what number is repeated in multiplication.

The exponent, on the other hand, indicates how many times the base is used as a factor in the multiplication.

• In our example, the exponent 2 tells us to multiply 0.5 by itself once, meaning \( 0.5 \times 0.5 \).

Recognizing and understanding these components will help simplify and solve many mathematical problems that involve powers.

Multiplication rule

The multiplication rule is vital when dealing with exponents, especially when squaring or generally raising numbers to a power. When exponents are involved, the operation often requires repeated multiplication of the base. For a base raised to the power of 2, also known as squaring, you only need to multiply the base by itself once.

• Example: For \( 0.5^{2} \), multiply 0.5 by itself: \( 0.5 \times 0.5 \).

This calculation follows the multiplication rule, which states:

• Multiply the base by itself as many times as specified by the exponent.

Following this rule ensures accurate results when solving equations involving exponents. It's a straightforward but crucial concept that reinforces how multiplication underpins the operation of exponentiation.