Problem 41 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The simplified expression is 2.7.

Step 1: Simplify the expression inside the parentheses

Start by simplifying what is inside the parentheses. We have \(5 + 3\). Calculating that gives us 8.

Step 2: Multiply the number outside the parentheses with result

Now, multiply 3 by the result from Step 1. So, \(3 \times 8 = 24\).

Step 3: Add the result to the remaining term in the numerator

Add 3 to the result from Step 2 to simplify the numerator. The numerator becomes \(3 + 24 = 27\).

Step 4: Calculate the denominator

Simplify the denominator \(3^2 + 1\). First, compute \(3^2 = 9\), then add 1, giving us 10.

Step 5: Divide the simplified numerator by the simplified denominator

Now divide the simplified numerator by the simplified denominator: \(27 \div 10 = 2.7\).

Key Concepts

Order of Operations

In algebra, the order in which you perform mathematical operations is crucial to obtaining the correct answer. This is known as the "order of operations," often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

When you apply this order, you ensure that all expressions are simplified correctly. In our exercise, the first task is to handle any operations inside parentheses. After tackling parentheses, we move to completing any multiplication or exponentiation within the problem. Only after these steps are all completed should you proceed to addition or subtraction. By following the order of operations, you achieve the correct final result every time.

Parentheses in Algebra

Parentheses are used in algebra to indicate which operations should be performed first. They act like a guide, telling you which parts of an expression are prioritized. When you see parentheses in a problem, start by solving the expressions contained within them. This builds the foundation for any subsequent calculations.

In the original exercise, the parentheses contain the expression \(5 + 3\). Solving this gives you 8. This result is crucial because it must be used before you can perform further operations involving the expression it is part of. Parentheses ensure accuracy and clarity, especially when multiple operations are at play, which is common in more complex algebraic expressions.

Numerator and Denominator Simplification

Simplifying a fraction involves a thorough simplification of both its numerator and denominator. Each needs to be broken down into its simplest form before performing the division.

For the numerator, after handling any operations inside the parentheses, as shown with \((5+3)\), you proceed by multiplying the result by any numeric values outside, in this case, the 3, giving \(3 \times 8 = 24\). Then, don't forget to add any other numbers present as per the order of operations, such as the 3 in the problem, resulting in a final numerator of 27.

For the denominator, start by working with any exponents, such as \(3^2\) which calculates to 9. Adding any remaining terms, like the 1 in this example, gives you 10 as the simplified denominator. Once both the numerator and denominator are simplified, they can be divided to obtain the simplest form of the expression, in this case, 2.7. Understanding these fractions can turn a seemingly complicated expression into something surprisingly simple.