Problem 53 - Intermediate Algebra

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: Simplify the Power

Start by simplifying the expression \(-\left(\frac{2}{3}\right)^{2}\). Calculate \(\left(\frac{2}{3}\right)^{2}\) which means \(\frac{2}{3}\times \frac{2}{3} = \frac{4}{9}\). Thus, \(-\left(\frac{2}{3}\right)^{2} = -\frac{4}{9}\).

Step 2: Simplify the Multiplication

Next, simplify the term \(5\left(\frac{2}{3}\right)\). Multiply 5 by \(\frac{2}{3}\) which results in \(\frac{10}{3}\). This gives us the expression: \(-\frac{4}{9} + \frac{10}{3} - 4\).

Step 3: Find Common Denominator

The fractions in the expression need a common denominator. For \(-\frac{4}{9}\), \(\frac{10}{3}\), and \(-4\), the common denominator is 9. Transform \(\frac{10}{3}\) to \(\frac{30}{9}\) (since \(10 \times 3 = 30\)) and \(-4\) to \(-\frac{36}{9}\) (since \(-4 \times 9 = -36\)).

Step 4: Simplify the Expression

Now add the fractions: \(-\frac{4}{9} + \frac{30}{9} - \frac{36}{9}\). Combine the fractions over the common denominator 9, calculating each step: - Combine the first two: \(-\frac{4}{9} + \frac{30}{9} = \frac{26}{9}\). - Then, \(\frac{26}{9} - \frac{36}{9} = \frac{-10}{9}\).

Step 5: Final Result

The simplified expression is \(-\frac{10}{9}\), which is already in its simplest form.

Key Concepts

Fractions Simplification

Simplifying fractions is about making them as straightforward as possible. To simplify a fraction, we find the greatest common divisor (GCD) of the numerator and the denominator. Then, we divide both by the GCD. For example, if we have \( \frac{10}{15} \), the GCD is 5. Dividing both, we get \( \frac{2}{3} \).

Remember that fractions can also appear as part of more complex expressions. It's essential to simplify each fraction independently before combining them.

It helps to look for common factors and cancel them out, which makes the fraction smaller and easier to manage. This skill becomes very useful later in algebraic manipulations and helps keep equations neat and tidy.

Common Denominator

When dealing with multiple fractions, we often need a common denominator, which is a shared denominator for all fractions involved. This is necessary to add or subtract fractions.

To find a common denominator, you can either find the least common multiple (LCM) of the denominators or multiply them together, which sometimes leads to larger numbers. For example, for \( \frac{1}{4} \) and \( \frac{1}{6} \), the LCM is 12, so we convert them to \( \frac{3}{12} \) and \( \frac{2}{12} \).

This approach standardizes the fractions, making it easy to perform any needed operations like addition or subtraction without altering their value. Remember, once you have a common denominator, only the numerators change when you perform these operations.

Polynomial Expressions

Polynomial expressions are mathematical phrases that can include numbers, variables, and exponents. They appear frequently in equations and need careful attention when simplifying.

A typical polynomial expression may look like \( 5x^2 + 3x - 4 \). Although our original problem doesn't include variables, the techniques involved are similar.

The goal is to combine like terms—terms that have the same variable raised to the same power. For instance, in simplifying \( 2x + 3x \), you combine them to get \( 5x \). Similarly, constants like \( -4 \) in the original example are combined after ensuring a common denominator when they appear in fractional form.

Simplifying expressions with both fractions and polynomials relies heavily on understanding each component and carefully executing arithmetic on them.