Problem 38 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \left(\frac{3 x^{2}}{56}\right)\left(\frac{3}{x}+\frac{5}{x}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{24x}{56} \).

Step 1: Distribute

To begin, distribute the first term \( \frac{3x^{2}}{56} \) to each term in the second parenthesis to obtain: \( \frac{3x^{2}}{56} \times \frac{3}{x} + \frac{3x^{2}}{56} \times \frac{5}{x} \).

Step 2: Simplify terms

Simplify each term separately. The \(x\) in the denominator and the \(x\) in the numerator will cancel out in each term: \( \frac{3 \times 3x^2}{56x} + \frac{3 \times 5x^2}{56x} \) which simplifies to: \( \frac{9x}{56} + \frac{15x}{56} \).

Step 3: Combine Like Terms

Combine the terms to get the final simplified expression: \( \frac{9x}{56} + \frac{15x}{56} \) results to \( \frac{24x}{56} \).

Key Concepts

Distributive Property

The distributive property is a key mathematical principle that allows you to multiply a term outside of parentheses by each term inside the parentheses. It simplifies expressions by "distributing" the multiplication over the addition or subtraction inside. This property is written in general form as:

• For addition: \( a(b + c) = ab + ac \)
• For subtraction: \( a(b - c) = ab - ac \)

When faced with an expression like \( \left( \frac{3x^{2}}{56} \right) \left( \frac{3}{x} + \frac{5}{x} \right) \), apply the distributive property by multiplying \( \frac{3x^{2}}{56} \) with each term inside the parentheses.
This results in two separate terms: \( \frac{3x^{2}}{56} \times \frac{3}{x} \) and \( \frac{3x^{2}}{56} \times \frac{5}{x} \).
By doing this, you break down the complex expression into simpler parts, making it easier to handle.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that contain the same variable raised to the same power. When you combine them, you add or subtract their coefficients. This is important for achieving the most simplified form of an expression.For example, after distributing and simplifying the expression \( \frac{9x}{56} + \frac{15x}{56} \), you notice that both terms are "like terms" because they have the same variables: "\(x\)".
To combine these, you simply add the coefficients (the numbers in front of the variables).So, \( \frac{9x}{56} + \frac{15x}{56} \) becomes \( \frac{24x}{56} \) by adding the coefficients 9 and 15. This ability to combine terms makes equations much simpler and more manageable.

Fractional Expressions

Fractional expressions involve fractions with variables in the numerator, the denominator, or both. Simplifying these expressions requires understanding how to manipulate fractions and variables simultaneously.When dealing with expressions like \( \frac{3x^{2}}{56} \times \frac{3}{x} \), you must consider both the numerical and variable aspects.

• Multiply fractions by multiplying the numerators together and the denominators together.
• Simplify by canceling out common factors in the numerator and the denominator.

In this example, the \( x \) in the numerator of \( 3x^{2} \) cancels with the \( x \) in the denominator, simplifying to \( \frac{9x}{56} \). Mastering fractional expressions allows you to perform arithmetic with algebraic fractions, greatly enhancing your math skills.