Problem 114 - College Algebra

Question

Use the order of operations to simplify each expression. \(\frac{10 \div 2+3 \cdot 4}{(12-3 \cdot 2)^{2}}\)

Step-by-Step Solution

Answer

The simplified value of the given expression is \(17/36\).

Step 1: Simplify the Numerator

Work out the division and multiplication first in the numerator, so \(10 \div 2+3 \cdot 4\) equals \(5+12\) which further simplifies to 17.

Step 2: Simplify the Denominator

Work out operation inside the brackets in the denominator. Start with multiplication, so \(12-3 \cdot 2\) equals \(12-6\) which further simplifies to 6. This value will be squared in the next step.

Step 3: Square the Denominator

Following the rules of BODMAS/BIDMAS/PEDMAS, the next step is to square the obtained value inside the brackets. So, \(6^{2}\) equals 36.

Step 4: Final Division

Finally, perform the division operation. Should be \(17/36\).

Key Concepts

Numerator Simplification

To simplify the numerator of a fraction, we need to follow the order of operations. This is often remembered using the acronym BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). In our exercise, the numerator is given as \(10 \div 2 + 3 \cdot 4\).

The operations here are division and multiplication, so we tackle them first. You begin by dividing 10 by 2, resulting in 5. Then, you multiply 3 by 4 to obtain 12.

Next, add the results: 5 plus 12 equals 17. This gives you the simplified form of the numerator.

• Division: \(10 \div 2 = 5\)
• Multiplication: \(3 \cdot 4 = 12\)
• Addition: \(5 + 12 = 17\)

With the numerator simplified, we can move on to focus attention on the denominator.

Denominator Simplification

Simplifying the denominator also requires close adherence to the order of operations. It begins with the expression \(12 - 3 \cdot 2\) inside the brackets. According to BODMAS, operations within brackets must be simplified initially.

The multiplication comes before subtraction, so you calculate \(3 \cdot 2 = 6\). Subtract this product from 12: \(12 - 6 = 6\).

Now that the operation within the brackets is done, the result 6 is ready for the subsequent step of squaring in the denominator.

• Inside Brackets: \(3 \cdot 2 = 6\)
• Subtraction: \(12 - 6 = 6\)

At this point, we have simplified the expression inside the bracket to 6, preparing it for squaring.

BODMAS/BIDMAS/PEDMAS

BODMAS, BIDMAS, and PEDMAS are acronyms that help remember the order of operations: Brackets, Orders (i.e., powers and roots), Division and Multiplication, Addition and Subtraction.

These rules are crucial for getting the correct result when simplifying mathematical expressions.

Both the numerator and the denominator need to be simplified according to these operations before any division takes place. Firstly, complete calculations inside brackets, then deal with powers, next address any multiplication or division, and finally perform any addition or subtraction.

For example, the expression contained a power in the denominator that needed addressing post simplification of the bracketed expression. The squared value of 6 following these orders results in \(6^2 = 36\).

• Brackets come first to simplify expressions within
• Following that, calculate any square, root, etc.
• Next come Division and Multiplication
• Finally, handle Addition and Subtraction

By staying true to BODMAS/BIDMAS/PEDMAS, errors in calculations are minimized, ensuring accuracy.

Division Operation

After simplifying both the numerator and the denominator, you conclude with the division operation. In our exercise, the result of our numerator is 17, and the squared denominator is 36.

Thus, the expression now stands as \(\frac{17}{36}\). This fraction represents the simplified form of the entire operation.

Division is straightforward once both sections of the fraction are simplified using BODMAS rules. Ensure all components are accounted for and all operations effectively executed to achieve the desired result.

• Numerator resolved to: 17
• Denominator resolved to: 36
• Final Fraction: \(\frac{17}{36}\)

Accurate division results in a precise, simplified fraction which completes this task.