Problem 42 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{7}{6}\).

Step 1: Simplify Inside the Parentheses

Start by simplifying the expression inside the parentheses, which is \(8 - 5\). This simplifies to \(3\). The expression now becomes \(3 + 6 \times 3\).

Step 2: Perform Multiplication

Next, multiply \(6\) by the result of the parentheses, \(3\). This gives you \(18\). The expression now is \(3 + 18\).

Step 3: Perform Addition in the Numerator

Add \(3\) and \(18\) to get \(21\). Thus, the numerator simplifies to \(21\).

Step 4: Evaluate the Exponent in the Denominator

Now, look at the denominator. Simplify \(4^{2}\), which equals \(16\). So, the denominator becomes \(16 + 2\).

Step 5: Perform Addition in the Denominator

Add \(16\) to \(2\) to get \(18\). Thus, the denominator simplifies to \(18\).

Step 6: Formulate the Simplified Fraction

Now substitute the simplified numerator and denominator into the fraction. This gives you \(\frac{21}{18}\).

Step 7: Simplify the Fraction

Look for the greatest common divisor (GCD) of \(21\) and \(18\). The GCD is \(3\). Divide both the numerator and the denominator by \(3\): \[\frac{21 \div 3}{18 \div 3} = \frac{7}{6}\]. This is the simplified form of the expression.

Key Concepts

Order of Operations

Whenever we tackle an algebraic expression, it's crucial to follow the order of operations. This set of rules helps us decide which calculation to perform first.
In mathematics, we often use the acronym PEMDAS to remember these rules:

• P: Parentheses - Complete operations inside parentheses first.
• E: Exponents - Solve these next, watch out for powers and roots.
• M/D: Multiplication and Division - Handle these from left to right as they appear.
• A/S: Addition and Subtraction - Finally, perform these operations from left to right.

Following this order ensures that everyone can work through a problem in the same way and come to the same answer. In our problem, you need to first handle parentheses, then exponents, followed by multiplication, and lastly, addition.

Fraction Reduction

Reducing fractions makes them simpler and easier to work with.
To reduce a fraction, divide both the numerator and the denominator by their greatest common factor.
For example, in the fraction \(\frac{21}{18}\), both numbers can be divided by 3. Doing this reduces the fraction to \(\frac{7}{6}\).

• Check for Common Factors: Start by looking for the largest number that evenly divides both the numerator and the denominator.
• Divide and Simplify: Once you have that number, divide both the top and bottom of the fraction to simplify it.

Reducing fractions is essential for making complex mathematical expressions more manageable and less intimidating.

Parentheses Handling

Dealing with parentheses might seem straightforward, but it's a critical step and always comes first in the order of operations.
Parentheses are used in math to group parts of an expression that need to be evaluated first. They can radically change what that expression means.
In our problem, the number within the parentheses, \(8 - 5\), simplifies to \(3\). This simple step sets the stage for the rest of the calculations.

• Evaluate First: Always start by solving any expressions within parentheses.
• Impact on Order: They help determine which parts of an expression to handle before anything else, influencing the overall result.

Respecting parentheses ensures accurate and error-free solutions.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that evenly divides two or more numbers.
Finding the GCD is essential when simplifying fractions.
For the expression \(\frac{21}{18}\), the GCD of 21 and 18 is 3. Dividing both by this value allows us to simplify the fraction to \(\frac{7}{6}\).

• Factorization: List out factors of each number.
• Identify the Largest Factor: Find the biggest number that appears in both lists.

Using the GCD makes numerical expressions much simpler, smoothing out calculations and reducing fractions efficiently.