Problem 36 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Then simplify the answer. $$ \frac{6 \cdot 4}{4+3^{2}-1} $$

Step-by-Step Solution

Answer

The simplified answer is 2.

Step 1: First deal with exponent

In the order of operations, exponents come before addition and subtraction.So, let's compute \(3^{2}\) first. This is equal to 9.

Step 2: Simplify the expression in the denominator

Now, look at the rest of the expression in the denominator which is \(4+9-1\). According to the order of operations, perform the addition and subtraction from left to right, yielding \(4+9=13\), and \(13-1=12\).

Step 3: Simplify the expression in the numerator

The numerator, \(6 \cdot 4\), is a simple multiplication operation and equals 24.

Step 4: Substitution

Now, substitute the numerator and denominator of the fraction with the values obtained i.e \(\frac{24}{12}\)

Step 5: Final Division

Lastly, divide 24 by 12, which simplifies to 2.

Key Concepts

Exponents

When dealing with exponents, they are a type of mathematical operation that represents repeated multiplication. In the expression provided, you encounter an exponent in the term \(3^2\). This means you multiply 3 by itself, resulting in 9. Exponents are crucial to manage early in any equation, due to the order of operations, commonly remembered by the acronym PEMDAS:

• P: Parentheses
• E: Exponents
• M/D: Multiplication or Division (from left to right)
• A/S: Addition or Subtraction (from left to right)

Understanding exponents helps simplify expressions before moving on to operations like addition or multiplication. They play a vital role in ensuring that the order of operations is followed correctly, allowing for accurate results.

Addition and Subtraction

Once you've dealt with any exponents, the next step in evaluating expressions often involves addition and subtraction. Consider the expression within the denominator \(4 + 3^2 - 1\), simplified to \(4 + 9 - 1\) after addressing the exponent.Following the order of operations, addition and subtraction should be handled from left to right as they appear:

• First, add 4 and 9 to get 13.
• Then, subtract 1 from 13, which equals 12.

When working through a problem, it helps to write each step out clearly, ensuring no mistakes are made. Keeping track of these operations ensures the correct simplification of components within your larger expression.

Multiplication and Division

The final steps often focus on multiplication and division, as these operations are crucial to simplifying expressions. In the original exercise, you see multiplication in the numerator with \(6 \cdot 4\), which equals 24.As shown in our solution, once the numerator and denominator are simplified, the division follows:

• Numerator: 24
• Denominator: 12
• Result: Divide 24 by 12 to achieve 2.

Remember, multiplication and division share the same level of priority in the order of operations. Solve them as they appear from left to right. Understanding this sequence is key to properly managing expressions, ensuring you reach the accurate final result.