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Chapter 5: Problem 2

Evaluate the expression: \(3(10+3) \div(8+5)+6 \cdot 4^2\)

Short Answer

Expert verified
The given expression can be simplified step-by-step using the order of operations (PEMDAS). The expression is \[3(10 + 3) \div (8 + 5) + 6 \cdot 4^2\] Upon simplification, the expression becomes: \[3(13) \div 13 + 6 \cdot 16\] By following PEMDAS, we get: \[39 \div 13 + 96\] Finally, we compute the value: \[3 + 96 = 99\] Hence, the simplified expression is \(\boxed{99}\).

Step by step solution

01

Solve expressions inside parentheses first.

We see that there are two sets of parentheses in the expression. We will solve each separately: \[3(10+3) \Rightarrow 3(13)\] and \[(8+5) \Rightarrow 13\]
02

Rewrite expression with simplified terms.

After solving the expressions inside parentheses, we rewrite the overall expression: \[3(13) \div 13 + 6 \cdot 4^2\]
03

Calculate exponents.

Our expression has one exponent term: \(4^2\). We will calculate its value: \[4^2 = 16\]
04

Rewrite expression with calculated exponent.

Now we rewrite the expression with the calculated exponent: \[3(13) \div 13 + 6 \cdot 16\]
05

Perform multiplication and division from left to right.

Now we perform multiplication and division from left to right: \[3 \cdot 13 = 39\] and \[6 \cdot 16 = 96\] Leaving us with: \[39 \div 13 + 96\] Next, we perform division: \[39 \div 13 = 3\]
06

Rewrite expression with simplified multiplication and division terms.

Now, we rewrite the expression with the simplified multiplication and division terms: \[3 + 96\]
07

Perform addition and subtraction from left to right.

Now, we perform addition from left to right: \[3 + 96 = 99\] The final result of the expression is 99.

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Key Concepts

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

Parentheses
Parentheses tell us to do specific calculations first. They set priorities in math problems, creating smaller, manageable expressions. Each expression inside parentheses should be simplified individually. In our example, two sets of parentheses exist: one for
  • \( 10 + 3 \)
  • \( 8 + 5 \)
First, calculate the sum inside each. For \( 10 + 3 \), you get 13. Similarly, solving \( 8 + 5 \) results in 13. Successfully managing these mini-tasks makes the rest of the problem easier.
Exponents
Exponents simplify repeated multiplication. When you see \( 4^2 \), it means \( 4 \) times itself. So \( 4^2 = 16 \). This calculation changes the layout of the problem. Replacing the exponent with its value maintains the original meaning. In our expression, this means updating the term \( 6 \cdot 4^2 \) to \( 6 \cdot 16 \). Remember, handle exponents after parentheses, but before other operations like multiplication or division.
Multiplication and Division
Both of these operations share the same priority level. Always move left to right, ensuring accuracy. With our expression, perform each step after organizing terms. Start with \( 3 \cdot 13 \) which equals 39. Then move to \( 6 \cdot 16 \), which calculates to 96. After completing multiplication, address division. Divide 39 by 13 to get 3. Each step's outcome simplifies the process, making further operations easier to tackle.
Addition and Subtraction
Once multiplication and division are completed, proceed to addition and subtraction. These operations form the final calculation stages as they鈥檙e the last priority. In our problem, having calculated up to \( 3 + 96 \), simply add the numbers to achieve the final outcome: \( 99 \). Clear understanding of this order ensures error-free calculations, ending with the correct result and offering a solid habit for handling complex expressions in future scenarios.

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Most popular questions from this chapter

Evaluate the expression: \(2 \cdot 4^2+3(5+1) \div 6=\)

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