Problem 49 - Intermediate Algebra

Question

Multiply. See Example 3 . $$\frac{2}{3}\left(3 s^{2}-9\right)$$

Step-by-Step Solution

Answer

The multiplied expression is \( 2s^2 - 6 \).

Step 1: Distribute the Fraction

We start by distributing the fraction \( \frac{2}{3} \) to each term inside the parentheses. The expression is \( \frac{2}{3}(3s^2 - 9) \), so distribute \( \frac{2}{3} \) to \( 3s^2 \) and \( -9 \).

Step 2: Multiply the First Term

Multiply \( \frac{2}{3} \) with the first term inside the parentheses, which is \( 3s^2 \). This results in \( \frac{2}{3} \times 3s^2 = 2s^2 \).

Step 3: Multiply the Second Term

Next, multiply \( \frac{2}{3} \) with the second term, \( -9 \). This results in \( \frac{2}{3} \times -9 = -6 \).

Step 4: Write the Final Expression

Combine the results from the multiplication to get the final expression: \( 2s^2 - 6 \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps in simplifying expressions and equations. This property states that if you have a number or term outside a set of parentheses multiplied by a sum or difference inside, you can distribute the multiplication to each term within the parentheses.
For example, when multiplying \( \frac{2}{3} \) by the terms inside the parentheses \((3s^2 - 9)\), you apply the distributive property. This involves:

• Multiplying \( \frac{2}{3} \) by \( 3s^2 \)
• Then multiplying \( \frac{2}{3} \) by \(-9\)

The results from these multiplications are then summed up, giving you a simplified expression. This method is especially useful when dealing with larger expressions and allows you to systematically simplify complex algebraic equations.

Fraction Multiplication

Fraction multiplication might seem tricky, but once you understand the process, it's quite straightforward. When multiplying fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together.
For instance, let's multiply \( \frac{2}{3} \) by \( 3s^2 \). To do this:

• Multiply the numerator 2 by 3s², giving \( 2 \times 3s^2 = 6s^2 \) before simplifying it to \( 2s^2 \) because \( \frac{6}{3} = 2 \).

The second multiplication is \( \frac{2}{3} \times -9 \), where you multiply:

• The numerator 2 by -9 resulting in -18, then divide by the denominator 3, which simplifies to -6.

This simplification process of fractions is crucial for simplifying more complex algebraic fractions into manageable pieces.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition, subtraction, multiplication, and division). In algebra, you are constantly working to simplify, solve, and transform these expressions into more usable forms.
In the exercise above, the algebraic expression \( \frac{2}{3}(3s^2 - 9) \) contains both a fraction and terms with variables. Understanding how to manage each component within the expression is key.

• The "3s^2" is a term with a variable \(s\) raised to the second power, signifying a polynomial term.
• The "-9" is a constant, a number without a variable, affecting the linear part of the expression.

When these are multiplied by a fractional coefficient, understanding the interaction between terms and operations is essential. Successfully manipulating these kinds of pieces in algebra paves the way for more advanced mathematical understanding and applications in solving equations and real-world problems.