Problem 26 - Algebra and Trigonometry

Question

Use properties of real numbers to write the expression without parentheses. \(\frac{4}{3}(-6 y)\)

Step-by-Step Solution

Answer

The simplified expression is \(-8y\).

Step 1: Understand the Problem

We need to simplify the expression \( \frac{4}{3}(-6y) \) by using properties of real numbers. Our goal is to remove the parentheses and simplify the expression.

Step 2: Apply the Distributive Property

The distributive property states that \( a(b + c) = ab + ac \). However, since we have a multiplication of two terms here, we multiply \( \frac{4}{3} \) by \(-6y\). This step simplifies the expression because there is only one term inside the parenthesis, not a sum or a difference.

Step 3: Multiply the Numbers

First, multiply the numerical coefficients: \( \frac{4}{3} \times -6 \). To do this, calculate \( 4 \times -6 = -24 \), and then divide by 3, resulting in \( \frac{-24}{3} \).

Step 4: Simplify the Multiplication

Simplify \( \frac{-24}{3} \) by dividing \(-24\) by \(3\), which results in \(-8\).

Step 5: Combine with Variable

Combine the simplified result \(-8\) with the variable \(y\), yielding the final simplified expression \(-8y\).

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra and a key property of real numbers that helps to simplify expressions and solve equations. It shows how multiplication interacts with addition or subtraction inside parentheses.

In its basic form, the distributive property is stated as:

• For any real numbers \(a\), \(b\), and \(c\), the property is expressed as: \[a(b + c) = ab + ac\]

This means if you multiply a number by a sum, you need to multiply it by each term inside the parentheses separately, then add the results.

However, in our specific example, the expression \(\frac{4}{3}(-6y)\), we are actually dealing with a single term inside the parentheses, which is a product with a negative coefficient. Here, direct multiplication applies, but understanding the distributive property ensures you know how terms can be expanded or reduced.

• When distributing a factor across a sum inside brackets, think of applying multiplication individually to all terms within the brackets.
• If only one term is present, such as in our exercise \(\frac{4}{3}(-6y)\), distribute by proceeding directly with the simple multiplication.

Simplifying Expressions Made Simple

Simplifying expressions involves breaking down a mathematical expression into its simplest or most reduced form, which is crucial for solving problems effectively. Here, our goal is to eliminate parentheses and simplify.

• Start by performing the required operations like multiplication or division.
• Recombine terms if necessary to reach a simpler form.

In the expression \(\frac{4}{3}(-6y)\), simplifying is straightforward. First, perform the operation inside the parentheses, if applicable, though in this case, it's a multiplication directly.

Next is carrying out the multiplication operation between the fractions or whole numbers and the variable if existing. Here, by effectively multiplying \(\frac{4}{3}\) with \(-6y\), numerical simplification leads to calculating \(\frac{-24}{3}\), which is then divided to yield \(-8y\).

Key steps in simplifying include:

• Remove any common factors if needed after multiplying.
• Combine like terms when applicable, focusing on coefficients and variables.

Removing parentheses once simplified enables the expression to be more manageable and easier to use in future computations.

Mastering the Multiplication of Fractions

When it comes to the multiplication of fractions, the process is often simpler than it appears and follows a clear and logical set of steps. To multiply fractions like \(\frac{4}{3}\) and a fraction formed with a whole number or another fraction, certain steps are followed.

• Multiply the numerators together: For instance, with \(\frac{4}{3} \times -6y\), consider the \(4\) and \(-6\), which results in \(-24\).
• Multiply the denominators: if the second number is a whole number, consider it as having a denominator of \(1\); thus, the denominator remains \(3\).

Following these rules, multiplying \(\frac{4}{3} \times -6\) follows through as \(\frac{4 \times -6}{3 \times 1} = \frac{-24}{3}\). This result is then simply divided to result in -8, which is then coupled with \(y\), the variable, resulting in \(-8y\).

A few key tips:

• Always simplify fractions to their lowest terms after multiplying for the cleanest solution.
• Remember, multiplying by a negative number will change the sign of the result.

These rules ensure accuracy and clarity in any problem involving multiplication of fractions, undoubtedly aiding in simplification tasks in algebra.