Problem 47 - Prealgebra

Question

The problems below review the material on exponents we have covered previously. Expand and simplify. $$(0.5)^{2}$$

Step-by-Step Solution

Answer

The expanded and simplified form is 0.25.

Step 1: Understanding the Problem

The task is to expand and simplify the exponential expression \((0.5)^2\). This means we need to find the square of 0.5.

Step 2: Rewrite the Exponent

The expression \((0.5)^2\) can be expanded by multiplying 0.5 by itself: \((0.5)^2 = 0.5 \times 0.5\).

Step 3: Perform the Multiplication

Compute the multiplication: \(0.5 \times 0.5 = 0.25\).

Step 4: Final Answer

The simplified form of \((0.5)^2\) is 0.25.

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical formulas into simpler, more manageable parts. When working with exponents, like \((0.5)^2\), simplifying means rewriting the expression in an equivalent but easier-to-understand form.
- Begin by recognizing the base and the exponent.- Here, the base is 0.5, and the exponent is 2.- The exponent tells us how many times to multiply the base by itself.
For our specific problem, this means rewriting \((0.5)^2\) as \(0.5 \times 0.5\). Simplifying doesn't change its value; it just makes the calculation more straightforward.
This skill is fundamental in mathematics education because it allows students to solve problems systematically and confidently.

Multiplication

Understanding multiplication is crucial for expanding expressions with exponents. When we say \((0.5)^2\), we are actually using multiplication in a repetitive manner.
- Multiplication is the process of adding a number to itself a certain number of times.- In this context, \(0.5 \times 0.5\) represents multiplying 0.5 by itself once.
Calculating this product, \(0.5 \times 0.5 = 0.25\), involves understanding how decimals work in multiplication. Make sure to align the numbers properly, even if they involve decimals, and then perform the standard multiplication steps.
This understanding plays a significant role in mathematics education, as it is a stepping stone to handling more complex calculations involving exponents and algebra.

Mathematics Education

Mathematics education focuses on building foundational skills like simplifying expressions and understanding multiplication. These are essential components of learning how to deal with exponents effectively.
- Education aims to ensure that students can break down problems into simpler parts.- By practicing problems like \((0.5)^2 = 0.5 \times 0.5 = 0.25\), students develop critical thinking and analytical skills.
The iterative process of expanding and simplifying helps in reinforcing the concept of exponents and prepares students for more advanced topics.
Good mathematics education encourages not only memorization of procedures but also comprehension of the underlying concepts, ensuring that students have a holistic understanding of mathematics.