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Chapter 3: Problem 15

Evaluate these expressions. a. \(-3 \cdot 8-5 \cdot 6\) b. \([-2-(-4)] \cdot 8-11\) c. \(7 \cdot 8+4 \cdot(-12)\) d. \(11-3 \cdot 9-2\)

Short Answer

Expert verified
a. -54 b. 5 c. 8 d. -18

Step by step solution

01

Evaluate Expression (a)

For the expression \(-3 \cdot 8 - 5 \cdot 6\), apply the order of operations. First, multiply each term. Calculate \(-3 \cdot 8 = -24\).Calculate \(-5 \cdot 6 = -30\).Now, summing them up gives \(-24 + (-30) = -54\).
02

Evaluate Expression (b)

For the expression \([-2 - (-4)] \cdot 8 - 11\), first simplify what's inside the brackets by resolving the subtraction: \(-2 - (-4) = -2 + 4 = 2\).Then, multiply the result by 8:\(2 \cdot 8 = 16\).Subtract 11 from this:\(16 - 11 = 5\).
03

Evaluate Expression (c)

For the expression \[7 \cdot 8 + 4 \cdot (-12)\], apply the same order of operations.First, calculate the products: \(7 \cdot 8 = 56\) and \(4 \cdot (-12) = -48\).Add these results: \(56 + (-48) = 8\).
04

Evaluate Expression (d)

For the expression \[11 - 3 \cdot 9 - 2\], multiply the term first:\(3 \cdot 9 = 27\).Subtract this product from 11: \(11 - 27 = -16\).Finally, subtract 2: \(-16 - 2 = -18\).

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Key Concepts

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

Multiplication
Multiplication is one of the fundamental operations in mathematics, used to combine groups of numbers in a fast and efficient manner. Understanding multiplication is key to transitioning into more complex math operations.
  • It involves calculating the product of two numbers or terms.
  • In the mathematical expressions we evaluated, like \(-3 \cdot 8\) and \(7 \cdot 8\), multiplication is performed before addition or subtraction, following the order of operations (PEMDAS/BODMAS).
Applying multiplication correctly requires attention to signs:
  • Multiplying two positive numbers results in a positive product.
  • Multiplying two negative numbers also results in a positive product, since "two negatives make a positive".
  • However, multiplying a positive and a negative number results in a negative product, as seen in \(4 \cdot (-12)\) which equals \(-48\).
Understanding these rules helps simplify order of operations, making problems easier to solve.
Subtraction
Subtraction is another core mathematical operation that involves taking away a number from another. This simple yet essential process is crucial across various mathematical contexts and problems.
  • Subtraction is used to find the difference between two numbers.
  • It often occurs after multiplication when evaluating expressions, as per the order of operations rules.
  • In expressions like \(16 - 11 = 5\) and \(-16 - 2 = -18\), subtraction helps achieve the final simplified result.
It鈥檚 important to pay attention to the order of terms in subtraction since changing the order can change the sign of the result. For instance, \(-2 - (-4)\) first requires resolving the double negative to get a positive before proceeding with the standard subtraction.
Negative Numbers
Negative numbers are numbers less than zero, represented with a minus sign (\(-\)). They can initially be confusing, but they follow their own set of predictable rules in arithmetic operations.
  • When subtracting a negative number, you effectively add the positive counterpart: \(-2 - (-4) = -2 + 4\).
  • Multiplying or dividing a negative by a positive number results in a negative outcome.
  • If both numbers are negative, the result will be positive, like \(4 \cdot (-12)\) results in a negative product.
Negative numbers are highly useful in real-world contexts, such as calculating debts, temperature differences, or direction in physics. Understanding their behaviour ensures accuracy in math problems.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They allow for general problem-solving through symbolic representation.
  • An expression often includes multiple operations and terms, such as \([ -2 - (-4) ] \cdot 8 - 11\).
  • The order of operations is crucial to simplify these expressions correctly, where multiplication and division are performed before addition and subtraction.
  • Variables often appear in algebra, and knowing how to manipulate expressions with them prepares students for solving more complex equations.
Algebraic expressions become a powerful tool once the basics are mastered, allowing for the solution of real-world problems involving unknowns and relationships between numbers.

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Most popular questions from this chapter

APPLICATION A long-distance telephone carrier charges \(\$ 1.38\) for international calls of 1 minute or less and \(\$ 0.36\) for each additional minute. a. Write a recursive routine using calculator lists to find the cost of a 7 -minute phone call. (A) b. Without graphing the sequence, give a verbal description of the graph showing the costs for calls that last whole numbers of minutes. Include in your description all the important values you need in order to draw the graph.

Positive multiples of 7 are generally listed as \(7,14,21,28, \ldots\). a. If 7 is the Ist multiple of 7 and 14 is the 2 nd multiple, then what is the 17th multiple? (a) b. How many multiples of 7 are between 100 and 200 ? (a) c. Compare the number of multiples of 7 between 100 and 200 with the number between 200 and 300 . Does the answer make sense? Do all intervals of 100 have this many multiples of 7? Explain. (a) d. Describe two different ways to generate a list containing multiples of 7 . (a)

At a family picnic, your cousin tells you that he always has a hard time remembering how to compute percents. Write him a note explaining what percent means. Use these problems as examples of how to solve the different types of percent problems, with an answer for each. a. 8 is \(15 \%\) of what number? b. \(15 \%\) of \(18.95\) is what number? c. What percent of 64 is 326 ? d. \(10 \%\) of what number is 40 ?

Solve these equations. Tell what action you take at each stage. a. \(144 x=12\) b. \(\frac{1}{6} x+2=8\)

Jo mows lawns after school. She finds that she can use the equation \(P=-300+15 \mathrm{~N}\) to calculate her profit. a. Give some possible real-world meanings for the numbers \(-300\) and 15 and the variable \(N\). b. Invent two questions related to this situation and then answer them. c. Solve the equation \(P=-300+15 N\) for the variable \(N\). d. What does the equation in \(7 \mathrm{c}\) tell you?
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