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Chapter 5: Problem 94

Use the order of operations to find the value of each expression. \((4-6)^{2}-(5-9)^{3}\)

Short Answer

Expert verified
The value of the expression \((4-6)^{2}-(5-9)^{3}\) is 68

Step by step solution

01

Solve Inside Parentheses

Carry out the operations inside the parentheses: (4-6) becomes -2 and (5-9) becomes -4
02

Apply Exponents

Raise -2 and -4 to the respective powers. \(-2^{2} = 4\) and \(-4^{3} = -64\)
03

Perform Subtraction

Subtract the two results from step 2: \(4-(-64) = 68\)

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Key Concepts

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

Understanding Exponents
Exponents are a way to express repeated multiplication of a number by itself. For instance, when you see something like \(x^{n}\), it means you multiply \(x\) by itself \(n\) times. In our problem, we have negative numbers being raised to exponents.

This means:
  • For \((-2)^{2}\), \(2\) is the exponent. So, you multiply \(-2 \, \times \, -2=4\) since multiplying two negative numbers results in a positive number.
  • For \((-4)^{3}\), \(3\) is the exponent. So, you multiply \(-4 \, \times \, -4 \, \times \, -4=-64\). Note that multiplying an odd number of negative numbers will result in a negative product.
This step is crucial in order of operations, and knowing how exponents work will help you tackle many math problems smoothly.
The Role of Parentheses
Parentheses are important because they tell you which operations to perform first when you鈥檙e solving math problems. Consider them as your 'first stop' in any math expression.

In our exercise, the expression inside the parentheses comes first:
  • You solve \(4 - 6\) to get \(-2\) and \(5 - 9\) to get \(-4\).
This step ensures that everything inside the parentheses is simplified before moving on to other operations like exponents and subtraction. Remember, ignoring parentheses will lead to incorrect results, so always deal with them early in the equation.
Simplifying with Subtraction
Subtraction in mathematics simply means taking one number away from another. It might look straightforward, but don鈥檛 overlook it!

In our expression, after calculating the exponents, we subtract the results: \(4 - (-64)\).

Why is it written like this? When you subtract a negative number, it becomes addition:
  • Subtracting \(-64\) is the same as adding \64\.
  • So, \(4 - (-64) = 4 + 64 = 68\).
Understanding how subtraction interacts with negative numbers is essential, especially when they鈥檙e involved with exponents and other operations. Knowing these rules will help you avoid mistakes in complex math problems.

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Most popular questions from this chapter

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