Problem 51 - Prealgebra

Question

Expand and simplify each of the following. $$\left(\frac{1}{2}\right)^{2}\left(\frac{3}{5}\right)^{2}$$

Step-by-Step Solution

Answer

The expanded and simplified expression is \( \frac{9}{100} \).

Step 1: Expand Each Term

We start by expanding the squared terms separately. First, expand \( \left( \frac{1}{2} \right)^2 \): \[ \left( \frac{1}{2} \right)^2 = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \].Next, expand \( \left( \frac{3}{5} \right)^2 \):\[ \left( \frac{3}{5} \right)^2 = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25} \].

Step 2: Multiply the Results

After expanding both terms, multiply the results:Multiply \( \frac{1}{4} \) by \( \frac{9}{25} \):\[ \frac{1}{4} \times \frac{9}{25} = \frac{1 \times 9}{4 \times 25} = \frac{9}{100} \].

Step 3: Final Answer

The expanded and simplified form of the expression is \( \frac{9}{100} \).

Key Concepts

Expanding Expressions

Expanding expressions involves taking a mathematical expression and rewriting it. This often helps you understand or simplify it.
When you see a term like \( (\frac{1}{2})^2 \), it means you're multiplying \( \frac{1}{2} \) by itself.
Specifically, expanding this squared term becomes:

• \( (\frac{1}{2})^2 = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)

Similarly, for \( (\frac{3}{5})^2 \), it expands to:

• \( (\frac{3}{5})^2 = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25} \)

By rewriting such terms using multiplication, you're breaking them down into simpler parts.
This makes it easier to handle complex expressions.

Multiplying Fractions

Multiplying fractions is straightforward if you follow the steps. You multiply the numerators together and the denominators together.
For example, if you have two fractions, \( a/b \) and \( c/d \), their multiplication is:

• \( \frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \)

In this specific example, after expanding, you need to multiply \( \frac{1}{4} \) by \( \frac{9}{25} \).
Here's how it's computed:

• \( \frac{1}{4} \times \frac{9}{25} = \frac{1 \times 9}{4 \times 25} = \frac{9}{100} \)

Ensure to multiply numerators to get the new numerator, and denominators to get the new denominator.
The result shows how the fractions have combined into a single simplified expression.

Squared Terms

Understanding squared terms is essential for working with various algebraic expressions.
The square of a number or a fraction means multiplying it by itself.
So, when you see \( (x)^2 \), understand that it implies \( x \times x \).For example:

• \( (\frac{1}{2})^2 \) indicates \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)
• \( (\frac{3}{5})^2 \) translates to \( \frac{3}{5} \times \frac{3}{5} = \frac{9}{25} \)

Squaring helps in manipulating expressions especially when solving equations.
Through expanding and simplifying, calculations become more manageable.