Problem 40 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(\frac{3+6(8-5)}{4^{2}+2}\)

Step-by-Step Solution

Answer

\(\frac{7}{6}\)

Step 1: Simplify Inside the Parentheses

Start by solving the expression inside the parentheses: \(8 - 5\). This simplifies to \(3\).

Step 2: Multiply the Result

Now, multiply the result from the first step by 6: \(6 \times 3\), which equals \(18\).

Step 3: Simplify the Numerator

Add \(3\) to the result from Step 2, which gives us \(3 + 18 = 21\).

Step 4: Simplify the Denominator

First, calculate \(4^2\), which is \(16\). Then add \(2\) to it, resulting in \(16 + 2 = 18\).

Step 5: Divide Numerator by Denominator

Divide the simplified numerator by the simplified denominator: \(\frac{21}{18}\). Simplify \(\frac{21}{18}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This results in \(\frac{7}{6}\).

Key Concepts

Order of Operations

The order of operations is a fundamental concept in mathematics used to solve expressions with different operations. It dictates the rules to follow when performing calculations in a complex expression. Here are some easy steps to follow:

• First, solve any expressions inside parentheses. In the exercise, we simplified \(8 - 5\) to get \(3\).
• Next, perform any multiplications or divisions from left to right. For instance, \(6\times3\) was calculated in Step 2.
• Lastly, handle any additions or subtractions from left to right. We added \(3 + 18\) in Step 3 for the numerator.

Remember to follow these steps carefully. Each operation needs to be done in the correct sequence to avoid mistakes. This set of guidelines ensures that everyone arriving at the same final result.

Fractions

Fractions represent parts of a whole and are an essential part of math. Simplifying fractions makes them easier to understand and work with. In our exercise, the expression was simplified to obtain the fraction \(\frac{21}{18}\). Here’s how to manage fractions properly:

• The top number of a fraction is called the numerator. It shows how many parts we have. In our case, the numerator was initially \(21\).
• The bottom number is called the denominator. It indicates how many parts the whole is divided into. For this exercise, the starting denominator was \(18\).
• If possible, always look to simplify your fraction by dividing both the numerator and the denominator by their greatest common divisor, as seen in Step 5.

Fractions might seem tricky at first, but practicing simplification can make you more comfortable handling them.

Greatest Common Divisor

The greatest common divisor (GCD) is key to simplifying fractions. It’s the largest number that can divide both the numerator and the denominator without leaving a remainder. Here’s a guide on finding and using the GCD:

• List the factors of each number. For \(21\) these are \(1, 3, 7, 21\), and for \(18\) they are \(1, 2, 3, 6, 9, 18\).
• Identify the highest number that appears in both lists. In our case, the GCD is \(3\).
• Use the GCD to simplify the fraction by dividing both the numerator and the denominator. \(\frac{21}{18}\) becomes \(\frac{7}{6}\).

Understanding and using the GCD is an effective way to make fractions more straightforward and easier to handle in your calculations.