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Chapter 1: Problem 38

\(181-\left(13 \cdot 3^{2}\right)\)

Short Answer

Expert verified
The result of the expression \(181-\left(13 \cdot 3^{2}\right)\) equals 64.

Step by step solution

01

Evaluate the Expression in Parentheses using Exponentiation

According to the order of operations, we should first compute any exponentiations within any parentheses. Thus, we must evaluate \(3^{2}\) which equals 9.
02

Compute the Multiplication

Next, we should compute any multiplication or division from left to right. Here, we have \(13 \cdot 9\), which equals 117.
03

Perform the Subtraction

Finally, the last operation we should perform is subtraction. So, we subtract 117 from 181, which gives us 64.

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Key Concepts

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

Exponentiation
When dealing with mathematical expressions, exponentiation comes into play whenever you see a power. Exponentiation involves multiplying a number by itself a certain number of times, indicated by the exponent. In our exercise, we have the expression \(3^{2}\). This means we multiply 3 by itself, resulting in 9.Here鈥檚 what happens:
  • Base: 3, which is the number we are multiplying.
  • Exponent: 2, indicating we multiply 3 by itself once (for a total of two 3s).
The result of \(3^{2}\) is 9, which simplifies our expression within the parentheses.
Parentheses
Parentheses are crucial in determining the order in which operations are executed in a mathematical expression. In our exercise, parentheses tell us to first solve everything inside them before moving on. Why does this matter?
  • They override the standard order of operations.
  • They help you group parts of an expression that should be computed first.
In the given expression \(181-\left(13 \cdot 3^{2}\right)\), we focus on operations inside \(\left( ... \right)\) and calculate them first. This approach prevents errors and ensures computations are accurate.
Multiplication
Once you've handled exponentiation inside the parentheses, you move on to multiplication. Multiplication tells us to combine groups of numbers or add a number to itself a certain number of times. In our example, we take the result from exponentiation and multiply it by 13. Here's how it breaks down:
  • The number to multiply: 9 (from exponentiation).
  • Multiplier: 13.
Multiplying 13 by 9, we get 117. Now we can replace the contents of the parentheses with this product, preparing for subtraction.
Subtraction
Subtraction is the final step in our order of operations. This operation involves taking away one value from another. Once we have completed the calculations inside parentheses and any necessary multiplication, we can proceed to the last step.In our equation,\(181-117\),
  • We start with the initial value: 181.
  • Subtract the value obtained from the parentheses: 117.
Ultimately, 181 minus 117 equals 64. This clean and clear approach through the order of operations ensures we get the correct result.

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