Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the expression. \(\left(\frac{4}{9}\right)^{2} \times 27 x\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{16x}{3}\).

Step 1: Simplify the Fraction

Simplify the expression by finding the square of the fraction provided. Square each part of the fraction separately: \[\left(\frac{4}{9}\right)^{2} = \frac{4^2}{9^2} = \frac{16}{81}\]

Step 2: Multiply by the Whole Number

Multiply the result from Step 1 by 27. Use the multiplication of fractions:\[\frac{16}{81} \times 27 = \frac{16 \times 27}{81}\]Simplify the fraction by dividing both the numerator and the denominator by 27:\[\frac{16 \times 27}{81} = \frac{16}{3}\]

Step 3: Include the Variable

Finally, include the variable \(x\) in the simplified expression:\[\frac{16}{3} \times x = \frac{16x}{3}\]

Key Concepts

Fractions in Algebraic Expressions

Fractions are a way of representing numbers through a numerator and a denominator. In the exercise, the fraction \(\frac{4}{9}\) is the base that needs to be squared. Understanding how to operate with fractions is crucial.

• Numerator: The top part of a fraction. It represents how many parts we have.
• Denominator: The bottom part of a fraction. It shows the total number of equal parts.

When you simplify fractions, you often need to reduce them to their simplest form to make calculations easier. Reducing includes dividing the numerator and the denominator by their greatest common divisor. Always keep an eye on whether a fraction can be simplified further after executing algebraic operations.

Exponentiation of Fractions

Exponentiation is a mathematical operation involving two numbers, the base, and the exponent. Applying exponentiation to a fraction means raising both the numerator and the denominator to the given power.
In our original exercise, this is expressed as: \[ \left(\frac{4}{9}\right)^{2} = \frac{4^{2}}{9^{2}} \]

• Squaring a number is multiplying the number by itself.
• So, \(4^{2} = 16\) and \(9^{2} = 81\), resulting in \(\frac{16}{81}\).

The process of applying exponentiation to each part of the fraction separately is crucial for maintaining the integrity of the operation. Remember that this principle applies for any power and not just squaring.

Variable Expressions and Their Importance

A variable is a symbol, often a letter, that represents a number. It's a cornerstone of algebra. In this expression, the letter \(x\) is the variable.
Variables allow us to create expressions that can represent many numbers depending on the value assigned to them. In algebraic operations:

• Variables are treated just like numbers while performing arithmetic operations.
• They can be combined, multiplied, or divided like any regular numeric value.

Including a variable in the final step of solving an algebraic expression allows the solution to be flexible and applicable to different scenarios by substituting different values for \(x\). This aspect of variable expressions is fundamental for solving equations and modeling real-world scenarios.

Step-by-Step Solutions for Better Understanding

Approaching a problem step-by-step is critical for simplifying algebraic expressions effectively. This method helps identify each action needed to get to the result and prevents mistakes.
In the provided exercise:

• We first solve for the fraction by squaring it (determining the fraction value).
• Next, we multiply the simplified fraction by a whole number to combine parts of the expression (focusing on arithmetic simplification).
• Finally, the variable \(x\) is multiplied into the expression (integrating the variable into the solution).

Taking a methodical approach allows for easier identification and correction of errors. By segmenting the problem into smaller, more manageable parts, you gain greater clarity and understanding.