Problem 53 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$-\left(\frac{2}{3}\right)^{2}+5\left(\frac{2}{3}\right)-4$$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{10}{9}\).

Step 1: Evaluate the Square

First, calculate the square of \( \frac{2}{3} \). The rule for squaring a fraction is to square both the numerator and the denominator: \( \left( \frac{2}{3} \right)^2 = \frac{2^2}{3^2} = \frac{4}{9} \).

Step 2: Apply the Negative Sign

Apply the negative sign to the square that was calculated in Step 1: \( -\left( \frac{4}{9} \right) = -\frac{4}{9} \).

Step 3: Calculate the Product

Next, calculate the product of \( 5 \) and \( \frac{2}{3} \). Multiply the integer by the numerator and keep the denominator: \( 5 \times \frac{2}{3} = \frac{10}{3} \).

Step 4: Express as a Common Denominator

Combine the fractions \( -\frac{4}{9} \), \( \frac{10}{3} \), and \(-4\) by expressing them with a common denominator. The least common denominator for 9 and 3 is 9. For \( -4 \), rewrite it as \( -\frac{36}{9} \).

Step 5: Combine Numerators

Convert \( \frac{10}{3} = \frac{10 \times 3}{3 \times 3} = \frac{30}{9} \). Now add \( -\frac{4}{9} + \frac{30}{9} - \frac{36}{9} \). Combine the numerators: \( -4 + 30 - 36 = -10 \).

Step 6: Simplify the Result

Express the result as a single fraction: \( -\frac{10}{9} \). This fraction is already in its simplest form.

Key Concepts

Fraction Operations

Understanding operations with fractions is crucial when simplifying numerical expressions involving them. Fractions have numerators and denominators. When performing calculations on fractions—like addition, subtraction, multiplication, and division—specific rules must be followed.

• To add or subtract fractions, they must have the same denominator. Only then can you combine the numerators while keeping the denominator the same.
• Multiplication involves multiplying numerators together and denominators together. For example, multiplying \( \frac{a}{b} \) and \( \frac{c}{d} \) results in \( \frac{ac}{bd} \).
• Division by a fraction involves multiplying by its reciprocal. The reciprocal of \( \frac{c}{d} \) is \( \frac{d}{c} \).

Mastering these fraction operations is vital for solving mathematical problems effectively, especially those involving numerical expressions.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. This ensures you're working with like terms instead of attempting to combine fractions with different bases.
When fractions have different denominators, follow these steps to find a common denominator:

• Identify the denominators of all fractions involved.
• Determine the least common multiple (LCM) of these denominators.
• Rewrite each fraction with this common denominator, adjusting the numerators accordingly.

For example, to combine \( \frac{4}{9} \), \( \frac{10}{3} \), and \(-4\), convert each to a common denominator of 9:

• \( \frac{10}{3} \) becomes \( \frac{30}{9} \).
• \(-4\) is expressed as \( -\frac{36}{9} \).

Once the fractions have a common denominator, you can easily add or subtract them by combining their numerators.

Order of Operations

The order of operations is a foundational concept in mathematics, dictating the sequence in which operations should be performed to correctly simplify expressions. Remember to follow the sequence known as PEMDAS:

• Parentheses: Complete operations inside parentheses first.
• Exponents: Evaluate powers and roots.
• Multiplication and Division: From left to right as they appear.
• Addition and Subtraction: Also from left to right.

Applying PEMDAS correctly ensures the accurate simplification of complex expressions. For instance, in the expression \(-\left(\frac{2}{3}\right)^2+5\left(\frac{2}{3}\right)-4\), handling the exponent first, followed by multiplication and then addition/subtraction, aligns with this order of operations.

Negative Numbers in Algebra

Dealing with negative numbers can be tricky, especially in algebraic expressions. However, several straightforward rules simplify these operations.

• A negative sign in front of a number or expression indicates its opposite.
• Two negative signs together make a positive. For instance, \(-(-a) = a\).
• Negative multiplied by positive (or vice versa) is negative, while negative multiplied by negative is positive.
• When subtracting, think of it as adding a negative. For example, \( 5 - 3 \) is the same as \( 5 + (-3) \).

In an expression like \(-\left(\frac{4}{9}\right)\), the negative sign changes the value from positive \( \frac{4}{9} \) to negative. Recognizing and applying these rules help manage negativity in expressions competently.