Problem 81 - Prealgebra

Question

Use the definition of exponents to simplify each expression. \((0.5)^{3}\)

Step-by-Step Solution

Answer

The expression \((0.5)^3\) simplifies to 0.125.

Step 1: Understand the Exponent

The expression \((0.5)^3\) means that the base \(0.5\) is multiplied by itself a total of three times. So we need to calculate \(0.5 \times 0.5 \times 0.5\).

Step 2: Multiply the Base with Itself the First Time

Let's first multiply the base \(0.5\) by itself once: \(0.5 \times 0.5 = 0.25\).

Step 3: Multiply the Result by the Base Again

Now use the result from Step 2 and multiply it by the base \(0.5\) one more time: \(0.25 \times 0.5 = 0.125\).

Step 4: Simplify the Expression

The simplified result of the expression \((0.5)^3\) is therefore 0.125.

Key Concepts

Simplifying Expressions

Simplifying expressions involving exponents is the process of expressing the calculation in its simplest form. For any given exponential expression like \((a)^n\),the 'simplifying' step involves writing down entire calculations expected by the powerand reducing them step-by-step.

• First, identify the base and exponents clearly.
• Translate the expression into a multiplication problem.
• Simplify sequentially through each multiplication step.

After understanding the number of times the base is multiplied,you simplify by calculating each multiplication one at a time until all have been executed.For example, in the expression \((0.5)^{3}\),this means we multiply the base 0.5 by itself three times.

Multiplication

Multiplication is at the core of working with exponents.The exponent tells us exactly how many times to multiply the base by itself.
When you multiply numbers, you combine equal groups together.In the expression \((0.5)^{3}:\)

• We first multiply \(0.5\) by \(0.5\), giving us 0.25.
• Then, we take that result, \(0.25\), and multiply by the base \(0.5\) again.
• The final multiplication gives us 0.125.

Understanding multiplication is essential as it acts as repeated addition but with a focus on grouping.
Each calculated multiplication builds off the last, eventually leading to the simplified final result.

Base and Exponent Concept

The base and exponent concept in mathematics is fundamental when dealing with powers.The **base** is the number that will be multiplied repeatedly.An **exponent** is a small number written slightly above and to the right of the base.It indicates how many times the base is used as a factor.
In\((0.5)^{3}\),the base is 0.5 and the exponent is 3.This indicates the base number 0.5 is multiplied by itself 3 times.

• The base: tells you "what number" you're working with.
• The exponent: tells you "how many times" to multiply the base by itself.

Understanding this concept allows you to interpret expressions and perform calculations accurately.