Problem 82 - Prealgebra

Question

Use the definition of exponents to simplify each expression. \((0.05)^{3}\)

Step-by-Step Solution

Answer

\((0.05)^{3} = 0.000125\).

Step 1: Understanding the Problem

We need to simplify the expression \((0.05)^{3}\) using the definition of exponents. Exponents indicate that a number (the base) is multiplied by itself a specific number of times (the exponent).

Step 2: Applying the Exponent Definition

Using the definition of exponents, we rewrite the expression \((0.05)^{3}\) as \(0.05 \times 0.05 \times 0.05\). This means multiplying 0.05 by itself three times.

Step 3: Calculating the Multiplication

First, we calculate \(0.05 \times 0.05\), which equals 0.0025. Then, we multiply the result by 0.05 again to get \(0.0025 \times 0.05 = 0.000125\).

Step 4: Conclusion

The simplified result of the expression \((0.05)^{3}\) is \(0.000125\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra and beyond, allowing us to work with more complex equations by breaking them down into easily manageable parts. In math, simplifying an expression often means rewriting it in a more concise or elegant form while preserving its value.
When dealing with expressions that include exponents, like \((0.05)^{3}\), we use the laws of exponents to help simplify.

• Identify the base and exponent in the expression.
• Rewrite the expression as repeated multiplication if needed.

By simplifying, you're not only getting closer to the answer but also gaining a better understanding of the underlying math involved. This process helps in improving problem-solving efficiency and accuracy.

Multiplication

Multiplication is one of the basic mathematical operations, which involves adding a number to itself a certain number of times. In terms of exponents, multiplication takes center stage since an exponent essentially tells us how many times to multiply a base by itself.
For example, in \((0.05)^{3}\), multiplication is the key operation, where 0.05 is multiplied by itself three times: \(0.05 \times 0.05 \times 0.05\).
Here's how you can think about it:

• Multiply any two of the base first, such as \(0.05 \times 0.05 = 0.0025\).
• Then, take the result and multiply it by the base one more time to get \(0.0025 \times 0.05 = 0.000125\).

Breaking down the multiplication into smaller steps can make complex calculations more straightforward and less error-prone.

Base and Exponent Concept

The base and exponent concept is integral to understanding powers in mathematics. In an expression like \((0.05)^{3}\), 0.05 is the base, and the number 3 is the exponent. The base is the number you multiply, while the exponent indicates how many times you multiply the base by itself.
This concept is handy as it allows for the simplification of expressions and easier handling of large numbers.

• Understanding it simplifies calculations by turning repeated multiplication into more dynamic operations.
• The exponent tells you the number of times the base appears as a factor in the multiplication process.

So, grasping the base and exponent relation is crucial for tackling exercises involving powers and can greatly enhance your mathematical fluency.