Problem 120 - College Algebra

Question

Use the order of operations to simplify each expression. \(\frac{12 \div 3 \cdot 5\left|2^{2}+3^{2}\right|}{7+3-6^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(-10\)

Step 1: Square the Numbers and Deal with the Absolute Value

The expression inside the absolute value symbols is \(2^{2} + 3^{2}\). Square both numbers to get \(4 + 9\), which is \(13\). The absolute value of \(13\) is \(13\). So, the expression becomes \(\frac{12 \div 3 \cdot 5 \cdot 13}{7 + 3 - 6^{2}}\)

Step 2: Deal with the Exponent

The next operation according to the order is exponentiation, which is \(6^{2}\). Squaring \(6\) gives \(36\). So, the expression becomes \(\frac{12 \div 3 \cdot 5 \cdot 13}{7 + 3 - 36}\)

Step 3: Perform Multiplication and Division from Left to Right in the Numerator

Multiplication and division are performed from left to right. So, the numerator becomes \(12 \div 3 \cdot 5 \cdot 13 = 4 \cdot 5 \cdot 13 = 20 \cdot 13 = 260\). Therefore, the expression is now \(\frac{260}{7+3-36}\)

Step 4: Perform Addition and Subtraction from Left to Right in the Denominator

The denominator becomes \(7 + 3 - 36 = 10 - 36 = -26\). Therefore, the expression is now \(\frac{260}{-26}\)

Step 5: Perform the Division

Finally, divide the numerator by the denominator to get the simplified expression: \(\frac{260}{-26} = -10\)

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on a number line. It's always a non-negative number, no matter if the value inside is positive or negative. In our expression, there is an absolute value function \( |2^2 + 3^2| \). Let's break down how we deal with this:

• First, you calculate the expressions inside. Here, \(2^2 + 3^2\) equals \(4 + 9 = 13\).
• The absolute value of \(13\) is still \(13\) because it's positive.

Effectively, we've turned the expression into \(12 \div 3 \cdot 5 \times 13\). This step is crucial because handling absolute values correctly ensures the right sequence of results in order of operations.

Exponentiation

Exponentiation involves raising a number to the power of another number. It can significantly change the magnitude of numbers in expressions, so it is performed early in the order of operations. In the given expression, we see \(6^2\) and need to evaluate this correctly:

• Raise 6 to the power of 2: \(6 \times 6 = 36\).

Now the expression in the denominator becomes \(7 + 3 - 36\). Tackling exponents first ensures simplifying the mathematical expression correctly. Remember, exponents are addressed before other operations like multiplication or addition.

Multiplication and Division

Multiplication and division come next after powers in the order of operations. They are performed together, moving from left to right across the expression. In the numerator of our fraction, we have operations ordered as \(12 \div 3 \cdot 5 \cdot 13\). Let's clarify how to address these:

• Start with the division: \(12 \div 3 = 4\).
• Proceed with multiplication: \(4 \cdot 5 = 20\).
• Continue to the next multiplication: \(20 \cdot 13 = 260\).

The correctly simplified form of the numerator is \(260\). Ensure you follow this left-to-right rule for multiplying and dividing, and you’ll maintain the integrity of the expression.

Addition and Subtraction

Addition and subtraction are the final steps in simplifying mathematical expressions using the order of operations. They are also tackled from left to right, helping wrap up the equation simplification process. Our focus here is on simplifying the denominator \(7 + 3 - 36\):

• First, do the addition: \(7 + 3 = 10\).
• Next, subtract: \(10 - 36 = -26\).

Handling these operations precisely results in the denominator being \(-26\). It's important to proceed sequentially to avoid mistakes, thus leading to the final division result of \(260 \div -26 = -10\). Consistency in dealing with these operations ensures complete and correct simplification.