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Chapter 2: Problem 13

For Exercises \(1-74,\) simplify. $$9^{2}-6 \cdot(-4) \div 8$$

Short Answer

Expert verified
Answer: The simplified result of the expression $9^2 - 6 \cdot(-4) \div 8$ is 84.

Step by step solution

01

Evaluate the Exponent

First, we need to evaluate the exponent. $$9^2 = 9 \cdot 9 = 81$$ So the expression becomes: $$81 - 6 \cdot(-4) \div 8$$
02

Perform Multiplication and Division (from left to right)

Now we need to perform the multiplication and division operations (from left to right). First, we have the multiplication: $$6\cdot(-4) = -24$$ So the expression becomes: $$81 - (-24) \div 8$$ Next, we perform the division: $$(-24)\div 8 = -3$$ So the expression becomes: $$81 - (-3)$$
03

Perform the Subtraction

Finally, we need to perform the subtraction operation. $$81 - (-3) = 81 + 3 = 84$$ Therefore, $$9^2 - 6 \cdot(-4) \div 8 = 84$$

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Key Concepts

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

Exponents
Exponents are a way to express repeated multiplication of a number by itself. When you see something like \(9^2\), it means that 9 is multiplied by itself two times. So, \(9^2 = 9 \cdot 9\). Exponents are one of the key operations you need to handle first when simplifying expressions, according to the order of operations rules. Remember that "PEMDAS" (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) helps us remember the order. In this example, we started by solving \(9^2\) which resulted in 81.
Multiplication and Division
Multiplication and division are operations that need to be handled next in the order of operations, and they should be performed from left to right.
  • First handle any multiplication in the expression: In the step-by-step solution, \(6 \cdot (-4)\) was calculated first, resulting in \(-24\).
  • Next, perform division: The result of the multiplication, \(-24\), was then divided by 8, which resulted in \(-3\).

Handling these operations correctly ensures that the correct value is used in subsequent steps when simplifying the expression.
Negative Numbers
Negative numbers can sometimes be tricky, especially when dealing with multiplication and division. It's important to remember:
  • Multiplying a positive number by a negative number results in a negative number. Thus, \(6 \cdot (-4) = -24\).
  • Dividing a negative number by a positive number results in a negative number, just like \((-24) \div 8 = -3\).

When subtracting a negative number, it is equivalent to adding its positive counterpart. This is why \(81 - (-3)\) is simplified to \(81 + 3\). Understanding how negative numbers interact with operations is crucial in accurately simplifying expressions.
Simplification
Simplification is the process of reducing an expression to its simplest form. We use the order of operations to ensure every part of an expression is calculated correctly step by step.
  • Start with exponents to simplify powers.
  • Next, perform any multiplication or division from left to right.
  • Finally, handle addition and subtraction.

In this exercise, after performing all operations in order, the expression \(81 - (-3)\) is simplified to \(81 + 3 = 84\). Simplification helps in making complex problems manageable and easier to understand by following a structured approach.

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Most popular questions from this chapter

Prep Exercise 8 For the following equation, complete the related division equation used to solve the equation. \(3 x=-15\) \(x=\) ____ solve and check. $$-1(-1)(-7) g=-63$$

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