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Chapter 1: Problem 33

Simplify. Show your steps. $$12 \div(4 \cdot 3)+5(9-6)^{2}$$

Short Answer

Expert verified
Question: Simplify the given expression and find the result: $$12 \div (4 \cdot 3) + 5(9 - 6)^{2}$$ Answer: 46

Step by step solution

01

Parentheses

Start by solving the expression inside the parentheses. $$12 \div (4 \cdot 3) + 5(9 - 6)^{2}$$ $$12 \div (12) + 5(3)^{2}$$
02

Exponents

Now, solve the exponent in the expression. $$12 \div 12 + 5(3)^{2}$$ $$12 \div 12 + 5(9)$$
03

Multiplication and Division (from left to right)

Perform the division first, as it appears before the multiplication. $$1 + 5(9)$$ Next, perform the multiplication. $$1 + 45$$
04

Addition and Subtraction (from left to right)

Finally, perform the addition. $$1 + 45 = 46$$ The simplified result is 46.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parentheses
When working with mathematical expressions, parentheses are used to group numbers or operations. This tells you which part of the expression to solve first. Think of parentheses as the highest priority task: whatever is inside them gets solved before anything else.

In the example exercise, we had the expression \(12 \div (4 \cdot 3) + 5(9 - 6)^{2}\) First, we evaluated the terms inside each set of parentheses:
  • For \((4 \cdot 3)\), we multiplied to get 12.
  • For \((9 - 6)\), we subtracted to get 3.
This simplification gives us the expression \(12 \div 12 + 5(3)^{2}\).

Always remember: Solve what's inside the parentheses first!
Exponents
Next in the order of operations is solving any exponents, which describe repeated multiplication. Exponents are sometimes called "powers" and are usually written with a smaller number raised to the right of the main number.

For instance, in our expression \(5(3)^{2}\),\((3)^{2}\) is the exponent. It means we multiply 3 by itself: \(3 \times 3 = 9\).By solving the exponent part \((3)^{2}\),we simplified it to 9, resulting in the expression:\(12 \div 12 + 5(9)\).

Exponents come right after parentheses and should be handled before moving on to multiplication, division, addition, or subtraction.
Multiplication and Division
After handling parentheses and exponents, it's time to focus on multiplication and division. These two operations are of equal priority, so we solve them as they appear from left to right in the expression.

In the given expression, \(12 \div 12 + 5(9)\),the division comes before multiplication, so we solve that first:
  • \(12 \div 12 = 1\)
This results in:\(1 + 5(9)\).

Next, we perform the multiplication:
  • \(5 \times 9 = 45\)
The expression is now simplified to \(1 + 45\).

Always perform multiplication and division from left to right.
Addition and Subtraction
Finally, we're left with addition and subtraction, which are handled last. Like multiplication and division, these operations are also of equal importance and should be executed from left to right.

In the case of the expression:\(1 + 45\)we simply add the numbers together:
  • \(1 + 45 = 46\)
This is our final result.

Remember, in any expression, always solve all additions and subtractions last, working from left to right.

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