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Chapter 3: Problem 62

Find the value of each of the following expressions. $$ \frac{-1(3+2)+5}{-1} $$

Short Answer

Expert verified
Answer: The value of the expression is 0.

Step by step solution

01

Evaluate the expression inside the parentheses

We first need to evaluate the expression inside the parentheses: $$ (3+2) $$ Since this is an addition operation, we simply add the two numbers together: $$ (3+2) = 5 $$
02

Multiply the result with the coefficient

Now, we have to multiply the result inside the parentheses with the coefficient in front of it, which is \(-1\): $$ -1(5) $$ Multiplying \(-1\) with \(5\) gives: $$ -1(5) = -5 $$
03

Add the other number to the result

After simplifying the multiplication, we need to add \(5\) to the result obtained in the previous step: $$ -5+5 $$ Adding \(-5\) and \(5\) gives: $$ -5+5 = 0 $$
04

Divide the result by the denominator

Lastly, we have to divide the result of the numerator by the denominator, which is \(-1\): $$ \frac{0}{-1} $$ Dividing \(0\) by \(-1\) gives: $$ \frac{0}{-1} = 0 $$ The value of the expression is \(0\).

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Key Concepts

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

Order of Operations
When evaluating mathematical expressions, it's essential to perform operations in a specific sequence to ensure accurate results. This sequence is known as the "Order of Operations". Remember the acronym **PEMDAS**, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

**Steps of Order of Operations:**
  • **Parentheses:** Simplify the expressions within parentheses first, just like in our exercise where the expression within the parentheses \(3+2\) was simplified to 5.
  • **Exponents:** Next, solve any exponents, though our exercise doesn鈥檛 include any.
  • **Multiplication and Division:** After parentheses and exponents, perform multiplication and division from left to right. Notice how we multiplied \(-1(5)\) next.
  • **Addition and Subtraction:** Finally, carry out addition and subtraction from left to right, as seen in step 3 where \(-5 + 5)\) simplified to 0.
Understanding this order is crucial because performing operations in a different sequence might lead to incorrect results. By following PEMDAS, we'll always arrive at the correct answer.
Integer Operations
Integer operations involve basic arithmetic鈥攁ddition, subtraction, multiplication, and division鈥攗sing whole numbers, also referred to as integers. These numbers can be positive, negative, or zero.

**Additions and Subtractions:**
  • When adding like signs, simply add the values and keep the sign. Example: \(3 + 2 = 5\).
  • When subtracting, think of the operation as adding a negative. For example, \(-5 + 5 = 0)\). Your numbers' signs matter.
**Multiplications and Divisions:**
  • For multiplication and division, two like signs (both positive or both negative) yield a positive result.
  • Two unlike signs (one positive and one negative) produce a negative result. That's why \(-1(5) = -5\).
Handling integer operations correctly is key, as any mistakes can change the outcome of the expression significantly.
Division by Negative Numbers
Dividing by negative numbers follows a straightforward rule, similar to multiplication with negative numbers. When you divide a number by a negative number, the quotient takes the opposite sign of the initial number being divided.

For example, consider dividing zero by a negative number as in our exercise: \(\frac{0}{-1}\). Because zero doesn't have a sign, the quotient remains zero. However, with positive or negative numbers:
  • A negative divided by a positive gives a negative quotient.
  • A positive divided by a negative yields a negative quotient, indicating a reversal of sign.
  • A negative divided by another negative results in a positive quotient, as two negatives cancel each other out.
While simple, understanding division by negative numbers adds to your toolbox for managing integers and ensures calculations are accurate, even when dealing with signs.

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Most popular questions from this chapter

Convert the numbers used in the following problems to scientific notation. The farthest object astronomers have been able to see (as of 1981 ) is a quasar named \(3 \mathrm{C} 427\). There seems to be a haze beyond this quasar that appears to mark the visual boundary of the universe. Quasar \(3 \mathrm{C} 427\) is at a distance of 110,000,000,000,000,000,000,000,000 meters from the earth.

Perform the following operations. $$ \left(2 \times 10^{-1}\right)\left(3 \times 10^{-5}\right) $$

Convert the numbers used in the following problems to scientific notation. Aluminum-26 has a half-life of 740,000 years.

Find the value of each of the following expressions. $$ \frac{3(4+1)-2(5)}{-2} $$

Convert the numbers used in the following problems to scientific notation. Near the surface of the earth, the speed of sound is 1195 feet per second.
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