Problem 12 - Intermediate Algebra

Question

In what order should the operations be performed to evaluate \(60-(-9)^{2}+5(-1) ?\)

Step-by-Step Solution

Answer

The order is: Evaluate exponents, multiplication, then perform subtraction and addition from left to right.

Step 1: Identify the Operations

Determine the types of operations present in the expression: subtraction, exponentiation, multiplication, and addition.

Step 2: Apply the Order of Operations

Recall the order of operations rule (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Step 3: Evaluate Exponentiation

Follow PEMDAS by evaluating the exponentiation first, handling \((-9)^{2}\). This results in \((-9) * (-9) = 81\).

Step 4: Evaluate Multiplication

Next, carry out the multiplication, which is 5 times -1. This results in \(5(-1) = -5\).

Step 5: Perform Subtraction and Addition

With the results of the previous operations, simplify the expression from left to right. Start with subtraction: \(60 - 81\) followed by addition: \(+5\).

Key Concepts

Exponentiation

Exponentiation is the process of raising a number to the power of another number. In the expression \(60-(-9)^{2}+5(-1)\), we encounter the exponentiation term \((-9)^{2}\). What this means is that the number -9 is multiplied by itself.
The important concept here is the handling of negative numbers with exponents. In particular, \((-9)^{2}\) implies multiplying -9 by -9, resulting in a positive value since the product of two negative numbers is positive.
Therefore, \((-9)^{2} = 81\). When evaluating expressions, always address exponents early; they come second in the order of operations after parentheses (PEMDAS/BODMAS). It’s crucial to apply exponentiation correctly to avoid errors in calculations.

Multiplication

Multiplication involves combining groups of equal sizes. In our given expression, \(5(-1)\) represents the multiplication of 5 with -1. To solve this:

• Simply multiply the absolute values: 5 times 1 equals 5
• The product of a positive and a negative number is negative

Hence, \(5(-1) = -5\).
As per the order of operations, multiplication is solved after parentheses and exponentiation but along with division as you proceed from left to right in the expression. Multiplying a negative number creates a change in the sign, which is vital to keep track of in solving complex equations.

Subtraction

Subtraction is the process of taking one number away from another to find the difference. In the expression, subtraction appears twice:

• First, solving \(60 - 81\) results from subtracting the result of the exponentiation from the initial number 60
• The second part involves dealing with the addition of a negative product, namely the result of \(5(-1) = -5\)

To correctly solve, following the order of operations is vital. Finish all multiplications and divisions before proceeding. Here, after calculating the relevant values, perform the subtraction from left to right in the sequence stated by the rules: - Calculate \(60 - 81\) which gives -21.- Add -5 as guided by the layout of the expression for the final result.Subtraction and addition are often grouped together as they both invert the value of the number being dealt with, signifying a change in value based on direction.