Problem 39 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(\frac{3+3(5+3)}{3^{2}+1}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{27}{10} \).

Step 1: Simplify the expression inside the parentheses

Start by simplifying the expression inside the parentheses:\[5 + 3 = 8\]

Step 2: Multiply

Now multiply the simplified result by 3:\[3 \times 8 = 24\]

Step 3: Add to the constant

Add the product from Step 2 to the constant in the numerator:\[3 + 24 = 27\]

Step 4: Simplify the denominator

Simplify the expression in the denominator:\[3^2 + 1 = 9 + 1 = 10\]

Step 5: Divide the simplified results

Divide the simplified numerator by the simplified denominator:\[\frac{27}{10}\]This fraction is already in its simplest form since 27 and 10 have no common factors other than 1.

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying expressions. It's a set of rules that determines the correct sequence to perform calculations. The order is often remembered by the acronym PEMDAS:

• Parentheses: Solve expressions inside parentheses first.
• Exponents: Then, calculate any exponents.
• Multiplication and Division: Perform these operations from left to right.
• Addition and Subtraction: Lastly, perform these from left to right.

In the provided exercise, the expression inside the parentheses is solved first. Then multiplication is conducted outside the parentheses. This systematic approach ensures the expression is simplified correctly.

Fractions

Fractions represent parts of a whole and are composed of a numerator and a denominator. In the expression's final step, the fraction \( \frac{27}{10} \) represents 27 parts of a whole divided into 10 parts.

Fractions simplify complex divisions and help in keeping calculations neat. When simplifying, fractions often reveal the simplest form of a problem, as seen in the example where the operation yielded the simplest form because no further common factors existed between the numerator and the denominator.

Numerator and Denominator

The numerator and denominator are core components of a fraction. They are separated by a line, forming a fraction that represents division. The numerator is the top number that indicates how many parts are being considered, while the denominator is the bottom number, indicating the total parts the whole is divided into.

In the expression \( \frac{27}{10} \), 27 is the numerator and tells us how many parts of the fraction we have. The number 10 is the denominator, informing us that the whole is divided into 10 equal sections. Understanding these components helps in simplifying and comparing fractions effectively.