Problem 62 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{3}{s^{2}}+\frac{7}{s t}+\frac{2}{t^{2}}}{\frac{2}{t^{2}}-\frac{5}{s t}-\frac{3}{s^{2}}} $$

Step-by-Step Solution

Answer

\( \frac{3t^2 + 7st + 2s^2}{2s^2 - 5st - 3t^2} \)

Step 1: Identify the Numerator and Denominator

In the given complex fraction, identify the numerator and the denominator. The numerator is \( \frac{3}{s^{2}} + \frac{7}{st} + \frac{2}{t^{2}} \) and the denominator is \( \frac{2}{t^{2}} - \frac{5}{st} - \frac{3}{s^{2}} \).

Step 2: Find a Common Denominator

Both the numerator and denominator of the complex fraction are expressions that contain fractions. For the numerator and the denominator, find a common denominator. The common denominator for both is \( s^2 t^2 \).

Step 3: Rewrite Each Term over the Common Denominator

Convert each term of the numerator and denominator to have the common denominator \( s^2 t^2 \):1. Numerator terms: \[ \frac{3}{s^2} = \frac{3t^2}{s^2 t^2}, \quad \frac{7}{st} = \frac{7st}{s^2 t^2}, \quad \frac{2}{t^2} = \frac{2s^2}{s^2 t^2} \]2. Denominator terms: \[ \frac{2}{t^2} = \frac{2s^2}{s^2 t^2}, \quad \frac{5}{st} = \frac{5st}{s^2 t^2}, \quad \frac{3}{s^2} = \frac{3t^2}{s^2 t^2} \]

Step 4: Combine Terms in Numerator and Denominator

Combine like terms in the rewritten numerator and denominator:1. Numerator: \[ \frac{3t^2 + 7st + 2s^2}{s^2 t^2} \]2. Denominator: \[ \frac{2s^2 - 5st - 3t^2}{s^2 t^2} \]

Step 5: Simplify the Complex Fraction

Divide the numerator by the denominator by multiplying by the reciprocal:\[ \frac{3t^2 + 7st + 2s^2}{s^2 t^2} \times \frac{s^2 t^2}{2s^2 - 5st - 3t^2} = \frac{3t^2 + 7st + 2s^2}{2s^2 - 5st - 3t^2} \]

Step 6: Conclusion

The simplified form of the complex fraction is: \[ \boxed{\frac{3t^2 + 7st + 2s^2}{2s^2 - 5st - 3t^2}} \]

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In simplifying complex fractions, algebra helps us understand the relationships between variables and constants. These symbols typically represent quantities that change and are often expressed in terms of equations and expressions.

In the given problem, the variables 's' and 't' are symbols used to represent quantities in the expressions. Algebra allows us to manipulate these expressions to achieve a simplified form. By applying algebraic rules, such as combining like terms and finding common denominators, complex fractions can be managed more efficiently. Understanding how these rules work is key to success in algebra.

Fraction Simplification

Fraction simplification involves reducing fractions to their simplest form. When dealing with complex fractions, it is essential that we simplify them to make calculations easier and to gain a clearer understanding of the solutions.

To simplify a complex fraction, you perform similar operations as you would for simple fractions. This means finding a common denominator and rewriting each term to consolidate them into a single expression with one numerator and one denominator. In our exercise, each part of the numerator and denominator was adjusted to share the common denominator of \( s^2 t^2 \).

Once this common denominator is in place, you can combine the terms more effectively, streamlining the fraction before further simplification.

Common Denominator

Finding a common denominator is essential when dealing with complex fractions. It allows us to combine multiple fractions into a single expression by ensuring they all have the same denominator.

For instance, in our example, each term in both the numerator and the denominator of the complex fraction needs to be rewritten with the common denominator \( s^2 t^2 \).

• The term \( \frac{3}{s^2} \) becomes \( \frac{3t^2}{s^2t^2} \).
• The term \( \frac{7}{st} \) becomes \( \frac{7st}{s^2t^2} \).
• And, the term \( \frac{2}{t^2} \) becomes \( \frac{2s^2}{s^2t^2} \).

After these adjustments, simplifying the complex fraction becomes much more manageable. The common denominator unifies the expression, allowing like terms to be summed algebraically.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and mathematical operations. In the context of complex fractions, mathematical expressions play a crucial role in framing problems and solutions.

The complex fraction in our problem contains expressions in both the numerator and denominator, involving addition and subtraction of fractional terms. To simplify these, we first look for a common path of simplification. This involves transforming each mathematical expression into forms that can be easily combined.

• Combining expressions involves performing arithmetic operations while observing algebraic rules.
• Expressions like \( 3t^2 + 7st + 2s^2 \) must be handled carefully to ensure terms are correctly combined.

By managing mathematical expressions with these steps, you manage to format the problem in a way that makes the solution clear and concise.