Chapter 1: Algebraic Fundamentals

1.34 What algebraic property guarantees that the equation in Problem 1.31 is true?

The sides of the equation in Problem 1.33 contain the same values; however, they are listed in a different order. The commutative property of multiplication states that re-ordering a set of real numbers multiplied together will not affect the product. (side note 1: Just like the associative property, the commutative property works for both addition and multiplication. If you've got a big list of numbers added together, you can add them in any order you want, and you'll get the same thing. In case you'd like to see visual evidence, here's Exhibit A: 1+9+6+2=6+1+2+9 10+6+2=7+2+9 16+2=9+9 18=18) (side note 2: "Product" is a fancy word for "what you get when you multiply things," like "sum" is a fancy way to say "what you get when you add things.")

Describe the number -2/9 by identifying the number sets to which it belongs. (1.10)

Because -2/9 is less than 0 (i.e., to the left of 0 on a number line), it is a negative number. It is a fraction, so by definition it is a rational number and, therefore, it is a real number and a complex number as well.

Describe the number 13 by identifying the number sets to which it belongs. (1.9)

Because 13 has no explicit decimal or fraction, it is an integer. All positive integers are also natural numbers and whole numbers. It is not evenly divisible by 2, so 13 is an odd number. In fact, 13 is not evenly divisible by any number other than 1 and 13, so it is a prime number. You can express 13 as a fraction (13/1), so 13 is a rational number. It follows, therefore, that 13 is also a real number and a complex number. In conclusion, 13 is odd, prime, a natural number, a whole number, an integer, a rational number, a real number, and a complex number. (side note: Any number divided by itself is 1, so 13/1 = 13

1.43 Identify the mathematical property that justifies the following statement and explain your answer: 3 x (5x6)x91x7)=(3x5)x(6x1)x7

Both sides of the equation contain the same numbers in the same order. The only difference between the sides of the equation is the way the numbers are grouped by the parentheses. Therefore, this statement is true by the associative property of multiplication, which states that the way a set of real numbers is grouped does not affect its product. (side note: If the numbers had been in a different order on either side of the equal sign, the commutative property would have come into play. The commutative property says "order doesn't matter," and the associative property says "it doesn't matter where you stick the parentheses.")

Is the number 0 even or odd? Positive or negative? Justify your answers. (1.3)

By definition, a number is even if there is no remainder when you divide it by 2. To determine whether 0 is an even number, divide it by 2: . (Note that 0 divided by any real number—except for 0—is equal to 0.) The result, 0, has no remainder, so 0 is an even number. However, 0 is neither positive nor negative. Positive numbers are defined as the real numbers greater than (but not equal to) 0, and negative numbers are defined as real numbers less than (but not equal to) 0, so 0 can be classified only as "nonpositive" or "nonnegative."

Simplify the expression: 6 x (-3). (1.14)

Choosing the sign to use when you multiply and divide numbers works very similarly to the method described in Problem 1.11 to eliminate double signs. When two numbers of the same sign are multiplied, the result is always positive. If, however, you multiply two numbers with different signs, the result is always negative. In this case, you are asked to multiply the numbers 6 and -3. Because one is positive and one is negative (that is, their signs are different), the result must be negative. 6 x (-3) = -18

Simplify the expression: -l5l-l-5l. (1.19)

If this problem had no absolute value bars and used parentheses instead, your approach would be entirely different. The expression -(5) - (-5) has the double sign "- -," which should be eliminated using the technique described in Problems 1.11-1.13. However, absolute value bars are treated differently than parentheses, so this expression technically does not contain double signs. Begin by evaluating the absolute values: l5l=5 and l-5l=5. -l5l-l-5l=-(5)-(+5) = -5-5 = -10 (side note 1: Well, it doesn't contain double signs YET. It will in just a moment.) (side note 2: See? There's the double sign. When l-5l turned into (+5), the negative sign in front of the absolute values didn't go away. In the next step, you eliminate the double sign "- +" to get -5 - 5.)

1.27 Simplify the expression: 3-12 divided by l-14l-5l

Like Problem 1.26, this fractional expression has, by definition, two implicit groups, the numerator and the denominator. However, it contains a second grouping symbol as well, absolute value bars. The absolute value expression is nested within the denominator, so simplify the innermost expression, first. 3-12 divided by l-14l-5 = 3-12 divided by 14-5Now simplify the numerator and denominator separately. 3-12 over 14-5 = -9/9Any number divided by itself equals 1, so =9/9= 1, but note that the numerator is negative. According to Problem 1.15, when numbers with different signs are divided, the result is negative. -9/9=-1

Identify the smallest positive prime number and justify your answer. (1.4)

A number is described as "prime" if it cannot be evenly divided by any number other than the number itself and 1. According to this definition, the number 8 is not prime, because the numbers 2 and 4 both divide evenly into 8. However, the numbers 2, 3, 5, 7, and 11 are prime, because none of those numbers is evenly divisible by a value other than the number itself and 1. Note that the number 1 is conspicuously absent from this list and is not a prime number. By definition, a prime number must be divisible by exactly two unique values, the number itself and the number 1. In the case of 1, those two values are equal and, therefore, not unique. Although this might seem a technicality, it excludes 1 from the set of prime numbers, so the smallest positive prime number is 2. (side note: Numbers, like 8, that aren't prime because they are divisible by too many things, are called "composite numbers.")

1.36 Describe the identity properties of addition and multiplication, including the role of the additive and multiplicative identities.

According to the identity property of addition, adding 0 (the additive identity) to any real number will not change the value of that number. Similarly, the multiplicative identity states that multiplying a real number by 1 (the multiplicative identity) doesn't change the value either. (side note: Don't over think this one—it's nothing you don't already know. If you multiply a number by 1 or add 0 to it, the number's IDENTITY doesn't change: 5 + 0 = 5 and 3x1=3.)

1.44 Identify the mathematical properties that justify the following statement and explain your answer: If 3 = x and x = y, then y = 3.

According to the transitive property, if 3 = x and x = y, then 3 = y. To rewrite 3 = y as y = 3 to match the given statement, you must apply the symmetric property. (side note: See Problem 1.42 for more information about the transitive property and Problem 1.37 for more info about the symmetric property)

List the following sets of numbers in order from smallest to largest: complex numbers, integers, irrational numbers, natural numbers, rational numbers, real numbers, and whole numbers. (1.8)

Although each of these sets is infinitely large, they are not the same size. The smallest set is the natural numbers, followed by the whole numbers, which is exactly one element larger than the natural numbers. Appending the negative integers to the whole numbers results in the next largest set, the integers. The set of rational numbers is significantly larger than the integers, and the set of irrational numbers is significantly larger than the set of rational numbers. The real numbers must be larger than the irrational numbers, because all irrational numbers are real numbers. The complex numbers are larger than the real numbers, as explained in Problem 1.7. Therefore, this is the correct order: natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, and complex numbers. (side note 1: According to Problem 1.1, the single element that the whole numbers contain and the natural numbers exclude is the number 0.) (side note 2: Any infinitely long decimal that has no pattern of repeating digits represents an irrational number. On the other hand, rational decimals either have to repeat or terminate. Because there are a lot more ways to write irrational numbers as decimals than there are to write rational numbers as decimals, there are a lot more irrational numbers than rational numbers)

1.39 Simplify the expression 3(2 - 7) (from Problem 1.38) once again, this time calculating the sum within the grouping symbols first. Verify that the result matches the answer produced by the distributive property in Problem 1.38.

Although the distributive property applies to this expression (as explained in Problem 1.38), simplifying the expression inside the grouping symbols first makes the problem significantly easier. 3(2-7)=3(-5) =-15 The expressions in Problems 1.38 and 1.39, though simplified differently, have the same value.

1.24 Simplify the expression: [19+(-11)]/2

Although this expression contains parentheses and brackets, the brackets are technically the only grouping symbols present; the parentheses surrounding -11 are there for notation purposes only. Simplify the expression inside the brackets first. [19+(-11)]/2=[19-11]/2 = 8 / 2 = 4 (side note: Double signs, like in the expression 19 + (-11), are ugly enough, but it's just too ugly to write the signs right next to each other like this: 19 + - 11. If you look back at Problems 1.11-1.13, you'll notice that the second signed number is always encased in parentheses if leaving them out would mean two signs are touching.)

Which is larger, the set of real numbers or the set of complex numbers? Explain your answer (1.7)

Combining the set of rational numbers together with the set of irrational numbers produces the set of real numbers. In other words, every real number must be either rational or irrational. The set of complex numbers is far larger than the set of real numbers, and the reasoning is simple: All real numbers are complex numbers as well. The set of complex numbers is larger than the set of real numbers in the same way that the set of human beings on Earth is larger than the set of men on Earth. All men are humans, but not all humans are necessarily men. Similarly, all real numbers are complex, but not all complex numbers are real. (side note: Complex numbers are discussed in more detail later in the book, in Problems 13.37- 13.44.)

1.20 Simplify the expression: l2l-l-7l+l-5l-9l.

Do not eliminate double signs in this expression until you have first addressed the absolute values. l2l-l-7l+l-5l-l9l=2-(+7)+(+5)-(+9)=2-7+5-9 Combine the signed numbers two at a time, working from left to right. Begin with 2 - 7 = -5. 2-7+5-9=-5+5-9 = 0-9 = -9 (side note: -5+5=0)

1.31 Simplify the expressions on each side of the following equation to verify that the sides of the equation are, in fact, equal. (3+9)+10=3+(9+10)

Each side of the equation contains a pair of terms added within grouping symbols. According to Problem 1.22, those expressions should be simplified first. Both sides of the equation have a value of 22 and are, therefore, equal.

1.26 Simplify the expression: 6+10 divided by 14-8.

Grouping symbols are not limited to parentheses, brackets, and braces. Though it contains none of the aforementioned elements, this fraction consists of two grouped expressions. Treat the numerator (6 + 10) and the denominator (14 - 8) as individual expressions and simplify them separately. 6+10 divided by 14-8 = 16/6 = 8/3 (side note: If you're not sure how 16/6 turned into 8/3, you divide the numbers in the top and bottom of the fraction by 2: 16/2=8 and 6/2=3. That process is called "simplifying" or "reducing" the fraction and is explained in Problems 2.11-2.17.)

Simplify the expression: (3)( -3)(4)(-4). (1.16)

Multiply the signed numbers in this expression together working from left to right. In this way, because you are multiplying only two numbers at a time, you can apply the technique described in Problem 1.14 to determine the sign of each result. The leftmost two numbers are 3 and -3; they have different signs, so multiplying them together results in a negative number: (3)( -3) = -9. (3)( -3)(4)( -4) = (-9)(4)( -4) Again multiply the two leftmost numbers. The signs of -9 and 4 are different, so the result is negative: (-9)(4) = -36. (-9)(4)( -4) = (-36)( -4) The remaining signed numbers are both negative; because the signs match, multiplying them together results in a positive number. (-36)( -4) = 144 (side note: There's no multiplication sign written between (3) and (-3), so how did you know to multiply them together? It's an "unwritten rule" of algebra. When two quantities are written next to one another and no sign separates them, multiplication is implied. That means things like 4(9), 10y, and xy are all multiplication problems.)

Describe the difference between the whole numbers and the natural numbers. (1.1)

Number theory dictates that the set of whole numbers and the set of natural numbers contain nearly all of the same members: {1, 2, 3, 4, 5, 6, ...}. The characteristic difference between the two is that the whole numbers also include the number 0. Therefore, the set of natural numbers is equivalent to the set of positive integers {1, 2, 3, 4, 5, ...}, whereas the set of whole numbers is equivalent to the set of nonnegative integers {0, 1, 2, 3, 4, 5, ...}. (side note 1: The natural numbers are also called the "counting numbers," because when you read them, it sounds like you're counting: 1, 2, 3, 4, 5, and so on. Most people don't start counting with 0.) (side note 2: So you get the whole numbers by taking the natural numbers and sticking 0 in there.)

Simplify the expression: l-10l-14. (1.18)

The absolute value of a negative number, in this case -10, is the opposite of the negative number: l-10l=10. l-10l-14=10-14=-4 (side note: Absolute values are simple when there's only one number inside. If the number inside is negative, make it positive and drop the absolute value bars. If the number's already positive, leave it alone and just drop the bars.)

1.40 Explain the additive inverse property and demonstrate it mathematically using a real number.

The additive inverse property states that adding any real number to its opposite results in the additive identity, 0. Consider the number 6; the sum of 6 and its opposite, -6, is 0: 6 + (-6) = 0. The property applies to negative numbers as well. Adding -3 to its opposite, +3, also results in 0: -3 + 3 = 0. (side note: Problem 1.36 explains why the additive identity is 0 and the multiplicative identity is 1.)

1.35 According to the associative properties of addition and multiplication, the manner in which values are grouped does not affect the value of the expression. However, Problems 1.22 and 1.23, which contain only addition and multiplication, prove that (3x7)+10 equal sign cross 3x(7+10). How is it possible that grouping the expressions differently changed their values, despite the guarantees of the associative properties?

The associative properties of addition and multiplication are separate and cannot be combined. In other words, you can apply the associative property of addition only when addition is the sole operation present, and you can apply the associative property of multiplication only when the numbers involved are multiplied. Neither associative property can be applied to the expression 3x7+10 because it contains both addition and multiplication.

1.38 According to the distributive property, if a, b, and c are real numbers, then a(b+c)=axb+axc. Apply the distributive property to simplify the expression 3(2 - 7).

The distributive property applies to expressions within grouping symbols that are multiplied by another term. Here, the entire expression (2 - 7) is multiplied by 3. The distributive property allows you to multiply each term within the parentheses by 3. 3(2-7)=3(2)+3(-7) Multiply 3(2) and 3(-7) before adding the terms together. According to the algebraic order of operations, multiplication within an expression should be completed before addition. For a more thorough investigation of this topic, see Problems 3.30-3.39. 3(2)+3(-7)=6+(-21) =6-21 =-15

List the two characteristics most commonly associated with a rational number. (1.5)

The fundamental characteristic of a rational number is that it can be expressed as a fraction, a quotient of two integers. Therefore, and are examples of rational numbers. Rational numbers expressed in decimal form feature either a terminating decimal (a finite, rather than infinite, number of values after the decimal point) or a repeating decimal (a pattern of digits that repeats infinitely). Consider the following decimal representations of rational numbers to better understand the concepts of terminating and repeating decimals. (some numbers and side note. Side note: Little bars like this are used to indicate which digits of a repeating decimal actually repeat. Sometimes, a few digits in front won't repeat, but the number is still rational. For example, is a rational number.)

What set of numbers consists of integers that are not natural numbers? What mathematical term best describes that set? (1.2)

The integers are numbers that contain no explicit fraction or decimal. Therefore, numbers such as 5, 0, and -6 are integers but 4.3 and are not. Thus, all integers belong to the set {..., -3, -2, -1, 0, 1, 2, 3, ...}. According to Problem 1.1, the set of natural numbers is {1, 2, 3, 4, 5, ...}. Remove the natural numbers from the set of integers to create the set described in this problem: {..., -4, -3, -2, -1, 0}. This set, which contains all of the negative integers and the number 0, is described as the "nonpositive numbers."

1.41 Explain the multiplicative inverse property and demonstrate it mathematically using a real number.

The multiplicative inverse property states that multiplying a number by its reciprocal results in the multiplicative identity, 1. For instance, if you multiply 2 by its reciprocal (1/2), the product is 1. 2x(1/2)=2/1x(1/2)=2/2=1 (side note 1:Problem 1.36 explains why the additive identity is 0 and the multiplicative identity is 1.) (side note 2: The reciprocal of an integer like 2 equals the fraction 1 divided by that number: 1/2. (So the reciprocal of -4 is -1/4and the reciprocal of 7 is 1/7.) The reciprocal of a fraction is the fraction you get by reversing the numerator and denominator. (So the reciprocal of 8/5 is 5/8 and the reciprocal of is -10/1, or just -10.) (side note 3: f multiplying or simplifying fractions makes you nauseated, don't worry. You'll find lots of practice in Problems 2.33- 2.37.)

1.30 Simplify the expression: absolute value of 4-l3-(-2)l over l6l+l1-(+4)l

The numerator and denominator both contain double signs within their innermost nested expressions. Begin simplifying there, and carefully work your way outward. For the moment, ignore the absolute value signs surrounding the entire fraction.absolute value of 4-l3-(-2)l over l6l+l1-(+4)l = absolute value of 4-l3+2)l over l6l+l1-4)l = absolute value of 4-l5l over l6l + (-3) = absolute value of 4-l5l over l6l-3 Evaluate l5l and l6l to continue simplifying. Now that the numerator and denominator each contain a single real number value, take the absolute value of the fraction that remains. l-1/3l=1/3 (side note: Leave the big, outside absolute value bars until the very end, after you have a single number on the top and bottom of the fraction.)

1.32 What algebraic property guarantees that the equation (3 + 9) + 10 = 3 + (9 + 10) from Problem 1.31 is true?

The only difference between the sides of the equation is the placement of the parentheses. According to the associative property of addition, if a set of numbers is added together, the manner in which they are grouped will not affect the total sum. (side note: There's also an associ-ative property for multiplication, which says that you can regroup numbers that are multiplied together and it won't change the answer. Here's an example: (4x5)x8=4x(5x8) 20x8=4x40 160=160)

1.23 Simplify the expression: 3x(7+10).

The only difference between this expression and Problem 1.22 is the placement of the parentheses. This time, the expression 7 + 10 is surrounded by grouping symbols and must be simplified first. 3x(7+10)=3x(17)=51 By comparing this solution to the solution for Problem 1.22, it is clear that the placement of the parentheses in the expression had a significant impact on the solution.

Simplify the expression: 4-l9l (1.17)

The straight lines surrounding 9 in this expression represent an absolute value. Evaluating the absolute value of a signed number is a trivial matter—simply make the signed number within the absolute value bars positive and then remove the bars from the expression. In this case, the number within the absolute value notation is already positive, so it remains unchanged. 4 - l 9 l = 4 - 9 You are left with two signed numbers to combine: +4 and -9. According to the technique described in Problem 1.11, combining $4 in assets with $9 in debt has a net result of $5 in debt: 4 - 9 = -5. (side note: Absolute value bars are the antidepressants of the mathematical world. They make everything inside positive. To say that more precisely, they take away the negative of the number inside. That means l-2l = 2. However, the moodaltering lines have no effect on positive numbers: l7l=7.)

1.37 Complete the following statement and explain your answer: According to the ______________ property, if a = b, then b = a.

The symmetric property guarantees that two equal quantities are still equal if written on opposite sides of the equal sign. In other words, if x = 5, then it is equally correct to state that 5 = x. (side note: So if you're as old as I am, then I am as old as you are. Hmmmm. Not very shocking or particularly groundbreaking.)

1.42 Complete the following statement and explain your answer: According to the transitive property, if a = b and b = c, then _______________.

The transitive property describes the relative equality of three quantities. Here, the quantity a is equal to the quantity b. In turn, b is equal to a third quantity, c. If b is equal to both a and c, it follows logically that a and c must also be equal. Therefore, the equation a = c correctly completes the statement.

1.33 Simplify the expressions on each side of the following equation to verify that the sides of the equation are, in fact, equal: 2x9x(-4)=(-4)x3x9

There are no grouping symbols present to indicate the order in which you should multiply the numbers on each side of the equation. Therefore, you should multiply the numbers from left to right, starting with 3x9 on the left side of the equation and (-4)x3 on the right.27x(-4)=-12x9 -108=-108 (side note: The rule stating that you should multiply a string of numbers from left to right is part of the order of operations. Problems 3.30-3.39 cover this in more detail.)

1.28 Simplify the expression: .

This expression consists of two separate absolute value expressions that are subtracted. The left fractional expression requires the most attention, so begin by simplifying it. absolute value of 3-3+6 over 1-8 -l-2l=absolute value of 1+6 over 1-8 - l2l = l7/-7l-l-2l = l-1l-l-2l Now that the fraction is in a more manageable form, determine both of the absolute values in the expression. l-1l-l-2l=1-(+2) =1-2 =-1 (side note: According to the end of Problem 1.27, when you divide a number and its opposite (like 7 and -7), you get -1.)

Simplify the expression: 16 + (-9). (1.11)

This expression contains adjacent or "double" signs, two signs next to one another. To simplify this expression, you must convert the double sign into a single sign. The method is simple: If the two signs in question are different, replace them with a single negative sign; if the signs are the same (whether both positive or both negative), replace them with a single positive sign. In this problem, the adjacent signs are different, "+ -," so you must replace them with a single negative sign: -. (some numbers and side notes. Side note 1: Some algebra books write positive and negative signs higher and smaller, like this: 16 + - 9. I'm sorry, but that's just weird. It's perfectly fine to turn that teeny floating sign into a regular sign: 16 + -9.) (side note 2: Think of it this way. If the two signs agree with each other (if they're both positive or both negative), then that's a good thing, a POSITIVE thing. On the other hand, when two signs can't agree with each other (one's positive and one's negative), then that's no good. That's NEGATIVE.)

Simplify the expression: -5 - (+6). (1.12)

This expression contains the adjacent signs "- +." As explained in Problem 1.11, the double sign must be rewritten as a single sign. Because the adjacent signs are different, they must be replaced with a single negative sign. -5 - (+6) = -5 - 6 To simplify the expression -5 - 6, or in fact any expression that contains signed numbers, think in terms of payments and debts. Every negative number represents money you owe, and every positive number represents money you've earned. In this analogy, -5 - 6 would be interpreted as a debt of $5 followed by a debt of $6, as both numbers are negative. Therefore, -5 - 6 = -11, a total debt of $11.

Simplify the expression: 4 - (-5) - (+10) (1.13)

This expression contains two sets of adjacent or "double" signs: "- -" between the numbers 4 and 5 and "- +" between the numbers 5 and 10. Replace like signs with a single + and unlike signs with a single -. 4 - (-5) - (+10) = 4 + 5 - 10 Simplify the expression from left to right, beginning with 4 + 5 = 9. 4 + 5 - 10 = 9 - 10 To simplify 9 - 10 using the payments and debts analogy from Problem 1.12, 9 represents $9 in cash and -10 represents $10 in debts. The net result would be a debt of $1, so 9 - 10 = -1. (side note: There's one other technique you can use to add and subtract signed numbers. If two numbers have different signs (like 9 and -10), then subtract them (10 - 9 = 1) and use the sign from the bigger number (10 > 9, so use the negative sign attached to the 10 to get -1 instead of 1). If the signs on the numbers are the same, then add the numbers together and use the shared sign. In other words, to simplify -12 - 4, add 12 and 4 to get 16 and then stick the shared negative sign out front: -16)

1.25 Simplify the expression: [30/(3x5)]-4.

This expression contains two sets of nested grouping symbols, brackets and parentheses. When one grouped expression is contained inside another, always simplify the innermost expression first and work outward from there. In this case, the parenthetical expression (3x5) should be simplified first. [30/(3x5)]-4=[30/15]-4A grouped expression still remains in the expression, so it must be simplified next. [30/15]-4=2-4 = -2 (side note: "Nested" means that one expression is inside the other one. In this case, (3x5) is nested inside the bracketed expression [30/(3x5)] because the expression inside parentheses is also inside the brackets. Nested expressions are like those egg-shaped Russian nesting dolls. You know the ones? When you open one of the dolls, there's another, smaller one inside?)

1.29 Simplify the expression: {[8-(3+l-1l)]+2}

This problem contains numerous nested expressions—braces that contain brackets that, in turn, contain parentheses that include an absolute value. Begin with the innermost of these, the absolute value expression {[8-(3+l-1l)]+2}={[8-(3+1)]+2}The innermost expression surrounded by grouping symbols is now (3 + 1), so simplify it next. {[8-(3+1)]+2}= {[8-4]+2} The bracketed expression is now the innermost group; simplify it next. {[8-4]+2}={4+2} = 6 (side note: Three if you don't count l-1l as a group (because it has only one number inside). Four if you do count it.)

1.21 Simplify the expression: l3+(-16)-(-9)l .

This problem contains the absolute value of an entire expression, not just a single number. In these cases, you cannot simply remove the negative signs from each term of the expression, but rather simplify the expression first and then take the absolute value of the result. To simplify the expression 3 + (-16) - (-9), you must eliminate the double signs and them combine the numbers one at a time, from left to right. l3+(-16)-(-9)l=l3-16+9l = l-13+9l = l-4l = 4

The irrational mathematical constant p is sometimes approximated with the fraction 22/7. Explain why that approximation cannot be the exact value of p. (1.6)

When expanded to millions, billions, and even trillions of decimal places, the digits in the decimal representation of p do not repeat in a discernable pattern. Because p is equal to a nonterminating, nonrepeating decimal, p is an irrational number, and irrational numbers cannot be expressed as fractions

1.22 Simplify the expression: (3x7)+10.

When portions of an expression are contained within grouping symbols—like parentheses (), brackets [], and braces {}—simplify those portions of the expression first, no matter where in the expression it occurs. In this expression, 3x7 is contained within parentheses, so multiply those numbers: 3x7=21. (3x7)+10=21+10 (side note: For now, the parentheses and other grouping symbols will tell you what pieces of a problem to simplify first. When parentheses aren't there to help, you have to apply something called the "order of operations," which is covered in Problems 3.30- 3.39.)

Simplify the expression: -16 / (-2). (1.15)

When signed numbers are divided, the sign of the result once again depends upon the signs of the numbers involved. If the numbers have the same sign, the result will be positive, and if the numbers have different signs, the result will be negative. In this case, both of the numbers in the expression, -16 and -2, have the same sign, so the result is positive: -16 / (-2) = 8. (side note: You could also write -16 / (-2) = +8, but you don't HAVE to write a + sign in front of a positive number. If a number has no sign in front of it, that means it's positive.