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Chapter 2: Problem 38

Simplify using the order of operations. $$64 \div(-4) \cdot 2 \div(-16)$$

Short Answer

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Step by step solution

01

Divide 64 by -4

First, perform the division: \(64 \div (-4)\). This simplifies to \(-16\).
02

Multiply the result by 2

Next, take the result from Step 1 and multiply it by 2: \(-16 \cdot 2\). This gives us \(-32\).
03

Divide the result by -16

Finally, divide the result from Step 2 by -16: \(-32 \div (-16)\). This simplifies to 2.

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Key Concepts

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

Division
Division is one of the four fundamental arithmetic operations. In this operation, one number (the dividend) is divided by another (the divisor) to give a result called the quotient. For example, in the expression \(64 \div (-4)\), 64 is the dividend and -4 is the divisor. The result is -16 since dividing a positive number by a negative number yields a negative quotient.

Here鈥檚 a quick reminder:
  • Dividing a positive number by a negative number results in a negative quotient, e.g., 64 梅 (-4) = -16.
  • Dividing a negative number by a positive number results in a negative quotient, e.g., -64 梅 4 = -16.
  • Dividing two negative numbers results in a positive quotient, e.g., -64 梅 (-4) = 16.

Once we have our quotient, we can move on to other operations such as multiplication.
Multiplication
Multiplication is another key arithmetic operation where one number is scaled by another. When you multiply a number by another, you are essentially adding the number to itself multiple times. For example, in the case of \(-16 \cdot 2\), we are doubling -16. This yields -32 because multiplying a negative number by a positive number results in a negative product.

It鈥檚 helpful to keep these rules in mind:
  • Multiplying a positive number by a positive number results in a positive product, e.g., 3 脳 4 = 12.
  • Multiplying a positive number by a negative number results in a negative product, e.g., 3 脳 (-4) = -12.
  • Multiplying a negative number by a negative number results in a positive product, e.g., (-3) 脳 (-4) = 12.

A solid understanding of these rules makes simplifying expressions much easier and can help prevent common mistakes.
Negative Numbers
Negative numbers are simply numbers less than zero. They are often used to represent quantities that are less than zero, such as debts or temperatures below freezing. In arithmetic operations with negative numbers, there are specific rules to follow to determine the sign of the result.

Here are the key points to keep in mind:
  • Adding a negative number is the same as subtracting the absolute value of that number, e.g., 5 + (-3) = 5 - 3 = 2.
  • Subtracting a negative number is the same as adding the absolute value of that number, e.g., 5 - (-3) = 5 + 3 = 8.
  • In multiplication and division, the product or quotient of two numbers with the same sign is positive, while the product or quotient of two numbers with different signs is negative.

Understanding these rules will make it easier to handle expressions involving negative numbers, like \(-32 \div (-16) = 2\). Since both numbers are negative, the quotient is positive.

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