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Chapter 1: Problem 47

Use the order of operations to simplify. $$(9-14)^{2}-(-3)(6)$$

Short Answer

Expert verified
The given expression is: \((9-14)^{2}-(-3)(6)\). Following the order of operations, first simplify inside the parentheses, then exponents, followed by multiplication, and finally addition. Simplifying step by step: \((-5)^{2}-(-3)(6)\) becomes \(25 + 18\), and the final simplified expression is \(43\).

Step by step solution

01

Parentheses

First, we will deal with the operation inside the parentheses: \(9 - 14 = -5\) Now the expression becomes: \((-5)^{2}-(-3)(6)\)
02

Exponents

Now we will deal with the exponent by squaring the value inside the parentheses: \((-5)^{2} = 25\) The expression becomes: \(25 - (-3)(6)\)
03

Multiplication

Next, we will deal with the multiplication: \(-(-3)(6) = 3 \times 6 = 18\) Our expression becomes: \(25 + 18\)
04

Addition

Finally, we will add the remaining numbers: \(25 + 18 = 43\) The simplified expression is: \(43\)

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Key Concepts

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

Parentheses
The first step in the order of operations is to handle any calculations inside parentheses. Parentheses are used to group parts of an expression that need to be evaluated prior to anything else. In our example, the expression inside the parentheses is \(9 - 14\). This operation should be performed first, according to the order of operations.

To solve \(9 - 14\), you subtract 14 from 9, which results in \(-5\). It's like borrowing when you're using money; if you have 9 dollars and you borrow 14, you'll be left with 5 dollars more to pay off, meaning you have -5 dollars.
  • Always evaluate everything inside the parentheses before moving on.
  • Multiple nested parentheses require you to work from the innermost set outwards.
This initial step transforms the expression into \((-5)^{2} - (-3)(6)\).
Exponents
Once the expressions inside the parentheses are resolved, the next part of the order of operations is dealing with exponents. Exponents are a way of expressing repeated multiplication of a number by itself. In our revised equation after simplifying the parentheses, we have \((-5)^{2}\).

The notation \((-5)^{2}\) means we multiply -5 by itself: \(-5 \times -5\). This results in 25. When multiplying two negative numbers, the negatives cancel each other out, leading to a positive number:
  • Negative times negative equals positive.
  • For instance, \(-5\times -5 = 25\).
After applying the exponent, the expression simplifies to \(25 - (-3)(6)\).
Multiplication
The next step in the order of operations involves dealing with multiplication. Multiplication is essentially repeated addition. In our simplified expression, \(25 - (-3)(6)\), we need to handle the multiplication of -3 and 6.

Multiply -3 by 6, which gives you -18. However, since the expression required a negative multiplication, and we must handle "minus a negative," the math simplifies to:
  • \(-3 \times 6 = -18\).
  • But \(-(-18)\) becomes \(+18\) because subtracting a negative is equivalent to adding a positive.
Therefore, the expression becomes \(25 + 18\).
Addition
Finally, after handling all other operations, we perform the addition. Our expression is now simplified to \(25 + 18\).

Addition is the process of calculating the total of two or more numbers. Simply put, you gather numbers to find the sum. Here, adding 25 and 18 is straightforward:
  • Add the ones: \(5 + 8 = 13\). Write down 3, carry over 1.
  • Add the tens: \(2 + 1 + 1 = 4\).
The sum is 43. Thus, the initial complex expression \((9-14)^{2} - (-3)(6)\) simplifies to 43, following the correct order of operations throughout each step.

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Most popular questions from this chapter

Rewrite each expression using the distributive property. Simplify if possible. $$-(6+1)$$

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