Problem 15 - Elementary Algebra

Question

For Problems \(1-20\), find the value of each numerical expression. For example, \(2^{4}=2 \cdot 2 \cdot 2 \cdot 2=16\). $$ \left(-\frac{3}{2}\right)^{2} $$

Step-by-Step Solution

Answer

\(\frac{9}{4}\)

Step 1: Understand the Expression

We are given the expression \( \left(-\frac{3}{2}\right)^{2} \). This means that we need to multiply \( -\frac{3}{2} \) by itself.

Step 2: Multiply the Base

To solve \( \left(-\frac{3}{2}\right)^{2} \), we perform the multiplication: \( -\frac{3}{2} \times -\frac{3}{2} \).

Step 3: Simplify the Multiplication

When multiplying fractions, multiply the numerators and the denominators separately. Thus, \((-3) \times (-3) = 9\) and \(2 \times 2 = 4\).

Step 4: Result of the Expression

The simplified fraction from the multiplication is \(\frac{9}{4}\). So, \(\left(-\frac{3}{2}\right)^{2} = \frac{9}{4}\).

Key Concepts

Numerical Expressions

Numerical expressions are combinations of numbers and operations like addition, subtraction, multiplication, and division. They are the building blocks of algebra and mathematics. When you see a numerical expression, it is your job to find the value by performing the operations in it. For example, when you see an expression like \(3 + (2 \/ 5) \times 4\), you need to do each operation in the correct order.

• Order of Operations: Use PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to guide you in solving these expressions.
• Grouping Symbols: Parentheses and brackets tell you to solve the operations inside them first.
• Exponents: Calculate the power expression before moving on to multiplication or division.

Start with any operations inside parentheses, then handle exponents, such as finding squares or cubes, before tackling multiplication, division, or the addition and subtraction steps.

Multiplying Fractions

When multiplying fractions, you multiply the top numbers (numerators) together and the bottom numbers (denominators) together. This rule simplifies the process.

• Multiply the numerators: This means taking the first fraction's numerator and multiplying it by the second fraction's numerator.
• Multiply the denominators: Take the first fraction's denominator and multiply it by the second fraction's denominator.
• Simplify: Look for any common factors between the numerator and the denominator of your answer. Simplify the fraction by dividing both by their greatest common divisor.

For instance, to multiply \(-\frac{3}{2}\) by itself, you would do the following:

• Calculate \((-3) \times (-3) = 9\)
• Calculate \(2 \times 2 = 4\)
• The product is \(\frac{9}{4}\)

This means \((-\frac{3}{2})^2" = \frac{9}{4}\) after squaring, which is a common practice in mathematical operations.

Squaring Negative Numbers

Squaring involves multiplying a number by itself. For negative numbers, it is important to note how the operation affects the sign. A common misunderstanding is that squaring a negative number results in a negative result. However, this is not the case.

• When you square a negative number: Multiply the number by itself, like \(-3 \times -3\).
• Negative sign rules: A negative multiplied by a negative equals a positive, which means that \(-3 \times -3 = 9\).
• Practical Example: In the expression \(\left(-\frac{3}{2}\right)^2\), multiplying \(-\frac{3}{2}\) by itself results in a positive value: \(\frac{9}{4}\).

Understanding this concept is essential for correctly solving expressions with powers or exponents involving negative numbers. Always keep in mind that the result of squaring any real number, regardless of its sign, will be non-negative.