Week 1 · Chapter

Chapter 1 — Real Numbers, Exponents & Algebraic Expressions

College Algebra Free sample edition

The module's primary reading · ~35–45 minutes · Objective 1 (classify real numbers, and simplify and evaluate algebraic expressions using the properties of real numbers, the order of operations, and the rules of integer exponents) · Read it first this week; the lecture, the tutorial, and the quiz all build on it.

In plain language

Algebra runs on a short list of rules that never change, and this week you learn them. You will sort numbers into families, such as the integers (the whole numbers and their negatives) and the irrational numbers (numbers whose decimals never stop and never repeat). You will learn the agreed order for doing arithmetic, so everyone who works a problem gets the same answer. You will learn the shortcuts for exponents (the small raised numbers that count repeated multiplication). Most of all, you will learn to check your own work, so you never have to wonder whether an answer is right.

Why you already care about this

Four friends finish dinner, and the check comes to 84 dollars. One friend adds a 20 percent tip to the whole check. Another adds a 20 percent tip to her own 21-dollar share and multiplies by four. A third types the numbers into a phone in a different order. All three should land on the same total, and they do. That is not luck. It happens because numbers obey rules that hold every single time.

You lean on those rules more often than you notice. You double a recipe that calls for 3 cups of flour and 2 cups of milk. You count how many teams fit in a tournament bracket. You split a bill, add a fee, or work out a discount. Each time, you trust that the order of the steps and the way you group the numbers will not change the answer. Sometimes that trust is right. Sometimes the order matters a great deal, and a calculator will happily give you a wrong answer if you type the steps in the wrong order.

This chapter shows you exactly which steps can be swapped and which cannot. It also gives you a set of shortcuts for exponents, built on a list of numbers you can check in your head: the powers of two.

By the end of this chapter you will be able to:

sort any real number into the natural numbers, whole numbers, integers, rational numbers, or irrational numbers;
evaluate an expression in the correct order, and tell −4² from (−4)²;
name the property of real numbers that justifies a step;
simplify a product, a quotient, or a power with the six exponent rules, and check the result by arithmetic;
simplify an expression by distributing and combining like terms, then evaluate it at a negative input.

Carry this sentence with you: every rule here can be checked — so check it.

Where we are

This week: the language of algebra — the kinds of real numbers, the order of operations, the properties of real numbers, the exponent rules, and simplifying and evaluating expressions.
How they connect: this chapter is the foundation, so there is no earlier week to lean on; every later week uses the rules you build here.
Where we're going next: Week 2, Linear Equations & Inequalities, uses these rules to find the number that makes a statement true.

This is the first week of the course, so nothing earlier is assumed beyond the arithmetic you already use: adding, subtracting, multiplying, dividing, and working with fractions. Next week you will move from simplifying an expression to solving an equation. Every step of that solving rests on the properties you meet in Section 3.

1. Sort the real numbers: name each number's family

Think about the numbers on a grocery receipt. There are counts, like 3 cans of beans. There is a negative, like a −2 dollar coupon. There are fractions of a pound, like 5⁄8 pound of cheese. There are prices in decimals, like 0.45 dollars. Mathematicians sort all of these into families. The families matter because some rules work only for some kinds of numbers.

Here are the families, from smallest to largest.

A natural number is one of the counting numbers 1, 2, 3, and so on; the set of natural numbers is written N.
A whole number is a natural number or zero, so the whole numbers are 0, 1, 2, 3, and so on.
An integer is a whole number or the opposite of a natural number, such as −3, 0, or 7; the set of integers is written Z.
A rational number is a number that can be written as a quotient of two integers with a nonzero denominator; its decimal form either stops or repeats, and the set is written Q.
An irrational number is a real number that cannot be written as a quotient of two integers; its decimal form goes on forever without repeating, as for √2 and π.
A real number is any number that is rational or irrational, which is every number that has a place on the number line; the set is written R.

The sets nest inside one another like boxes. Every natural number is an integer. Every integer is a rational number, because you can write 7 as 7⁄1. Every rational number is a real number. So N sits inside Z, Z sits inside Q, and Q sits inside R. The irrational numbers are the real numbers that are not rational. They fill the gaps on the number line that fractions miss.

One more term before the example. The principal square root of a nonnegative number is the nonnegative number whose square equals it, written with the √ sign, so √16 = 4.

Worked example — classify eight numbers. Sort 7, −3, 0, 5⁄8, √16, √2, 0.45, and π. For each one, ask: can it be written as a fraction of two integers?

7 is a counting number, so 7 is a natural number. It is also whole, an integer, rational, and real.
−3 is the opposite of a counting number, so −3 is an integer. It is not whole, because whole numbers start at 0.
0 is not a counting number, but 0 is a whole number. It is also an integer.
5⁄8 is already a fraction of two integers, so 5⁄8 is rational.
√16 looks like a radical, but work it out first. 4² = 16, so √16 = 4. That makes √16 a natural number.
√2 ≈ 1.4142, and its decimal never stops or repeats, so √2 is irrational.
0.45 is forty-five hundredths. Write it as 0.45 = 45⁄100 = 9⁄20, so 0.45 is rational.
π is the ratio of a circle's distance around to its distance across. Its decimal never stops or repeats, so π is irrational.

In words: six of the eight numbers are rational, and only √2 and π are irrational.

Misconception check. Students often think any number written with a √ sign is irrational. The sign is tempting because it looks unfinished. But √16 = 4, because 4² = 16, so √16 is a plain natural number. The cure is to evaluate first and classify second. A square root is irrational only when the number under the sign is not a perfect square, as with √2.

Honesty note: some courses count 0 as a natural number. This course does not, and no graded question will turn on that choice.

Memory hook: evaluate first — then name the family.

2. Follow the order of operations: two left-to-right passes

Type 2 + 3 · 4 into a basic four-function calculator, one key at a time. Many of those calculators add first and show 20. A scientific calculator multiplies first and shows 14. Both cannot be right. Mathematicians settled the question long ago with an agreement, and every graphing tool you will use follows it.

The order of operations is the agreed sequence for evaluating an expression: grouping symbols first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

Notice the two left-to-right pairs. Multiplication and division are partners; you do whichever comes first as you read left to right. Addition and subtraction are partners in the same way. So 2 + 3 · 4 is 2 + 12 = 14, and the scientific calculator is right.

Two words describe every power. The base is the number or letter that is multiplied by itself; in 2⁵ the base is 2. The exponent is the raised number that says how many factors of the base to multiply when it is a whole number; in 2⁵ the exponent is 5.

Worked example — evaluate 2 + 3 · 4². Go through the order one level at a time.

Grouping symbols: there are none.
Exponents: 4² = 4 · 4 = 16. The expression becomes 2 + 3 · 16.
Multiplication: 3 · 16 = 48. The expression becomes 2 + 48.
Addition: 2 + 48 = 50.

In words: 2 + 3 · 4² equals 50. If you multiplied before squaring, you would get 2 + 12² = 2 + 144 = 146 — a different number from the same digits.

The order of operations also settles a small but famous question: what does a minus sign in front of a power mean?

A negated power is an expression such as −4², where the exponent applies only to 4 and the minus sign is applied afterward, so −4² = −16.

Worked example — a negated power versus a negative base. Compare −4² with (−4)².

In −4², the exponent binds first. Square the 4: 4² = 16. Then apply the minus sign: −4² = −(4²) = −16.
In (−4)², the parentheses make −4 the base. Multiply it by itself: (−4)² = (−4)(−4) = 16. A negative times a negative is positive.

The same thing happens with a fourth power: −2⁴ = −16, but (−2)⁴ = 16. In words: the parentheses decide whether the minus sign is part of the base.

Misconception check. Students often read −2⁴ as 16, because "a negative to an even power is positive" sounds like a rule. That rule is true, but only when the negative is the base. In −2⁴ the base is 2, not −2, so you compute 2⁴ = 16 and then take the opposite, which gives −16. The cure is to ask one question before you compute: is the minus sign inside the parentheses with the number?

In practice (your tools). Open the free Desmos graphing calculator in a browser and type 2+3*4^2 on an empty line; it shows 50. Type -4^2 on the next line and it shows −16; type (-4)^2 and it shows 16. Desmos follows the order of operations exactly, so it is a quick check on your hand work, not a replacement for it.

Memory hook: grouping, exponents — then two left-to-right passes.

3. Use the properties of real numbers: why the shortcuts work

Back to dinner. Three friends order meals costing 14, 9, and 11 dollars. One friend adds 14 + 9 first and then adds 11. Another adds 9 + 11 first. Both get 34 dollars, and nobody argues. That calm agreement rests on properties of real numbers. A property is a rule that holds for every real number, so you can use it without checking each case.

The commutative property is the rule that changing the order of two numbers in a sum or in a product does not change the result: a + b = b + a and ab = ba.
The associative property is the rule that changing how three numbers are grouped in a sum or in a product does not change the result: (a + b) + c = a + (b + c) and (ab)c = a(bc).
The distributive property is the rule that a factor multiplies every term of a sum: a(b + c) = ab + ac; read right to left, it is what lets you combine like terms.
The identity property is the rule that adding 0 or multiplying by 1 leaves a number unchanged: a + 0 = a and a · 1 = a.
The inverse property is the rule that a number plus its opposite is 0, a + (−a) = 0, and a nonzero number times its reciprocal is 1, a · (1⁄a) = 1.

A reciprocal of a nonzero number a is the number 1⁄a, which multiplies a to give 1; the reciprocal of 4 is 1⁄4.

Worked example — three everyday checks. Each step below names the property that allows it.

The shared bill (associative). Add the meals two ways. (14 + 9) + 11 = 23 + 11 = 34. Then 14 + (9 + 11) = 14 + 20 = 34. The grouping changed and the total did not.

The doubled recipe (distributive). A recipe uses 3 cups of flour and 2 cups of milk. Doubling it means 2(3 + 2) cups of ingredients in all. Add first: 2(5) = 10 cups. Or distribute: 2 · 3 + 2 · 2 = 6 + 4 = 10 cups. Same 10 cups either way.

The tip (distributive, read right to left). A meal costs 40 dollars and you leave a 20 percent tip. The total is 40 + 0.2 · 40 dollars. Factor out the 40: 40(1 + 0.2) = 40(1.2) = 48 dollars. Check it the long way: 0.2 · 40 = 8, and 40 + 8 = 48 dollars. Multiplying by 1.2 adds the tip in one step.

In words: regrouping a sum, distributing a factor, and factoring one out all leave the value alone.

The distributive property combines like terms

The distributive property does one more job, and it is the one you will use most. Like terms are terms that have the same variable raised to the same exponent, such as 5x and −6x, which the distributive property lets you add. Here a variable is a letter, such as x, that stands for a number that can change or that is not yet known.

To combine 3x + 5x, read the distributive property from right to left: 3x + 5x = (3 + 5)x = 8x. You are counting x's, the way 3 apples plus 5 apples makes 8 apples. But 3x + 5x² cannot be combined, because x and x² are different powers.

Misconception check. Students often assume subtraction and division are commutative, because addition and multiplication are. They are not. Try it with numbers: 10 − 4 = 6, but 4 − 10 = −6. And 12 ⁄ 3 = 4, but 3 ⁄ 12 = 1⁄4. The cure is to rewrite a subtraction as adding the opposite, 10 − 4 = 10 + (−4), before you swap anything.

Pause and reflect — choose one. Pick either prompt; they count the same.

(a) Think of a time you split a cost with other people; which property of real numbers made the split come out the same no matter who added first?
(b) Think of a recipe or a project you scaled up; where did you multiply every ingredient or part by the same number, and what would have gone wrong if you had multiplied only one?

Memory hook: the factor reaches every term, not only the first one.

4. Build the exponent rules on the powers of two

Picture a single-elimination tournament. Each round cuts the field in half. So a bracket for 2 teams has one round, a bracket for 4 teams has two rounds, and a bracket for 8 teams has three. The team counts are powers of two, and they make the exponent rules easy to check.

Here is the list you will use all week: the powers of two from 2⁰ through 2¹⁰.

The powers of two, 2⁰ through 2¹⁰, with each exponent and its exact value
Exponent	0	1	2	3	4	5	6	7	8	9	10
Value	1	2	4	8	16	32	64	128	256	512	1024

The list has 11 values, from 1 up to 2¹⁰ = 1024, and they add to 2047. Each value is double the one before it. That doubling is exactly what an exponent counts.

Because every value is known, you can test each rule below by plain arithmetic. That is the whole point: you never have to take an exponent rule on faith.

The product rule: when two powers with the same base are multiplied, keep the base and add the exponents: aᵐ · aⁿ = aᵐ⁺ⁿ.
The quotient rule: when a power of a nonzero base is divided by another power of the same base, keep the base and subtract the exponents: aᵐ ⁄ aⁿ = aᵐ⁻ⁿ for a ≠ 0.
The power rule: when a power is raised to another power, keep the base and multiply the exponents: (aᵐ)ⁿ = aᵐⁿ.
The power of a product: when a product is raised to a power, every factor is raised to that power: (ab)ⁿ = aⁿbⁿ.
The zero exponent: any nonzero base raised to the exponent 0 equals 1: a⁰ = 1 for a ≠ 0.
The negative exponent: an exponent that calls for a reciprocal, a⁻ⁿ = 1⁄aⁿ for a ≠ 0; it never makes the value itself negative.

Worked example — test every rule on the table. Use the powers of two for each check.

Product rule. 2³ · 2⁴ = 2³⁺⁴ = 2⁷. Check it: 2³ = 8 and 2⁴ = 16, and 8 × 16 = 128. The table says 2⁷ = 128. It matches.
Quotient rule. 2¹⁰ ⁄ 2⁷ = 2¹⁰⁻⁷ = 2³. Check it: 1024 ⁄ 128 = 8, and the table says 2³ = 8.
Power rule. (2²)³ = 2⁶ = 64, because the exponents multiply: 2 × 3 = 6. Check it: 2² = 4, and 4³ = 4 · 4 · 4 = 64.
Power of a product. (2x)³ = 2³x³ = 8x³. Check it at x = 3. The left side is (2 · 3)³ = 6³ = 216. The right side is 8 · 3³ = 8 · 27 = 216.
Zero exponent. Divide a power by itself: 2³ ⁄ 2³ = 2³⁻³ = 2⁰. But 8 ⁄ 8 = 1. So 2⁰ = 1.
Negative exponent. 2³ ⁄ 2⁵ = 2³⁻⁵ = 2⁻². But 8 ⁄ 32 = 1⁄4. So 2⁻² = 1⁄2² = 1⁄4.

In words: each rule is a shortcut for counting factors, and each one passed its arithmetic check. Back in the bracket picture, 2³ regions of 2⁴ teams each hold 2⁷ = 128 teams. That is 128 teams versus 16 in one region, an 8-fold change, so an organizer has 8 times as many teams to schedule.

The zero and negative rules are not new inventions. They are what the quotient rule forces. Read the table from right to left: each step divides by 2. From 2¹ = 2, one more halving gives 2⁰ = 1, and one more gives 2⁻¹ = 1⁄2.

Misconception check. Students often write x² · x³ = x⁶, multiplying the exponents. It is tempting because 2 and 3 make 6. But multiplying powers adds exponents: x² · x³ means (x · x)(x · x · x), which is five factors of x, so x² · x³ = x⁵. Multiplying exponents is the power rule, for a power raised to a power, such as (x²)³ = x⁶. Check with the table: 2² · 2³ = 4 · 8 = 32 = 2⁵, not 2⁶ = 64.

Misconception check — a negative exponent is not a negative number. Students often read 2⁻² as −4. The minus sign in the exponent looks like it should make the answer negative. It does not. A negative exponent means "take the reciprocal," so 2⁻² = 1⁄4, a small positive number. The cure is to say the rule out loud every time: negative exponent, flip it.

Caveat: the zero-exponent rule needs a nonzero base. The expression 0⁰ is left undefined in this course, and you will not need it.

Memory hook: multiply powers, add exponents; power of a power, multiply them.

5. Simplify and evaluate: distribute, combine, then substitute

Picture a campus print shop. A poster costs 12 dollars, and the shop adds a flat 4-dollar handling fee to every order. The cost of an order of t posters is the expression 12t + 4. Before you can use an expression like that, you need two skills. You need to tidy it up, which is simplifying. And you need to put a number in for the letter, which is evaluating.

An algebraic expression is a combination of numbers, variables, and operation signs with no equals sign, such as 3x² − 2x + 5. To evaluate an expression is to find its value by substituting a number for each variable and then following the order of operations.

For the print shop, an order of 3 posters costs 12(3) + 4 = 36 + 4 = 40 dollars. That number helps a club decide whether 3 posters fit its budget.

Worked example — distribute a negative across a sum. Simplify 5x − 2(3x − 4).

Treat the subtraction as multiplying by −2. The −2 must reach both terms inside the parentheses.
Distribute: −2 · 3x = −6x, and −2 · (−4) = +8. So 5x − 2(3x − 4) = 5x − 6x + 8.
Combine like terms: 5x − 6x = (5 − 6)x = −1x, which is written −x.
The simplified form is −x + 8.

Check it by substitution at x = 2. The original gives 5(2) − 2(6 − 4) = 10 − 2(2) = 10 − 4 = 6. The simplified form gives −2 + 8 = 6. The two match, so the simplification is right.

Misconception check. Students often write 5x − 2(3x − 4) = 5x − 6x − 8. The minus sign in front of the 2 seems to belong only to the first term. But the −2 multiplies both terms, and −2 times −4 is +8. Dropping that second sign is one of the most common errors in all of algebra. It is a slip, not a deep misunderstanding, and the cure is a check by substitution every time.

Worked example — evaluate at a negative input. Evaluate 3x² − 2x + 5 at x = −2.

Replace every x with (−2), in parentheses: 3(−2)² − 2(−2) + 5.
Exponent first: (−2)² = 4. This is the step to watch.
Multiply: 3 · 4 = 12, and −2(−2) = +4. The expression becomes 12 + 4 + 5.
Add from left to right: 12 + 4 = 16, and 16 + 5 = 21.

In words: 3x² − 2x + 5 equals 21 when x = −2.

The same expression gives a full table if you try more inputs. At x = −2, −1, 0, 1, 2, and 3 the outputs are 21, 10, 5, 6, 13, and 26. The smallest is 5, at x = 0, and the largest is 26, at x = 3. You will meet this table again when it becomes a function.

Misconception check — the coefficient and the exponent. Students often evaluate 3x² as (3x)², squaring the 3 along with the x. At x = −2 that gives (−6)² = 36 instead of 3 · 4 = 12. The exponent belongs only to x. The coefficient 3, the number that multiplies the variable, is applied after the square.

In practice (your tools). In Desmos, type 3(-2)^2-2(-2)+5 on an empty line; it shows 21. To see the whole table at once, choose the plus button at the top left of the expression list and pick "table." Enter −2, −1, 0, 1, 2, and 3 in the first column, then click the second column's heading and type 3x_1^2-2x_1+5. That column fills in 21, 10, 5, 6, 13, and 26.

Limitation: a substitution check catches most errors, but a single lucky value can hide a mistake. It is the same habit the four friends used at dinner: two routes, one total. If a check passes, it is strong evidence. If you want more certainty, check a second value.

Memory hook: parentheses on every negative — no exceptions.

Key terms

natural number — One of the counting numbers 1, 2, 3, and so on; the set of natural numbers is written N. (first used: 1. Sort the real numbers: name each number's family)
whole number — A natural number or zero, so the whole numbers are 0, 1, 2, 3, and so on. (first used: 1. Sort the real numbers: name each number's family)
integer — A whole number or the opposite of a natural number, such as −3, 0, or 7; the set of integers is written Z. (first used: 1. Sort the real numbers: name each number's family)
rational number — A number that can be written as a quotient of two integers with a nonzero denominator; its decimal form either stops or repeats, and the set is written Q. (first used: 1. Sort the real numbers: name each number's family)
irrational number — A real number that cannot be written as a quotient of two integers; its decimal form goes on forever without repeating, as for the square root of 2 and pi. (first used: 1. Sort the real numbers: name each number's family)
real number — Any number that is rational or irrational, which is every number that has a place on the number line; the set is written R. (first used: 1. Sort the real numbers: name each number's family)
principal square root — For a nonnegative number, the nonnegative number whose square equals it, so the principal square root of 16 is 4. (first used: 1. Sort the real numbers: name each number's family)
order of operations — The agreed sequence for evaluating an expression: grouping symbols first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. (first used: 2. Follow the order of operations: two left-to-right passes)
base — In a power, the number or letter that is multiplied by itself, such as the 2 in two to the fifth power. (first used: 2. Follow the order of operations: two left-to-right passes)
exponent — In a power, the raised number that says how many factors of the base to multiply when it is a whole number, such as the 5 in two to the fifth power. (first used: 2. Follow the order of operations: two left-to-right passes)
negated power — An expression in which the exponent applies only to the number and the minus sign is applied afterward, so negative four squared means the opposite of 16. (first used: 2. Follow the order of operations: two left-to-right passes)
commutative property — The rule that changing the order of two numbers in a sum or in a product does not change the result. (first used: 3. Use the properties of real numbers: why the shortcuts work)
associative property — The rule that changing how three numbers are grouped in a sum or in a product does not change the result. (first used: 3. Use the properties of real numbers: why the shortcuts work)
distributive property — The rule that a factor multiplies every term of a sum; read in reverse, it is what lets you combine like terms. (first used: 3. Use the properties of real numbers: why the shortcuts work)
identity property — The rule that adding 0 or multiplying by 1 leaves a number unchanged. (first used: 3. Use the properties of real numbers: why the shortcuts work)
inverse property — The rule that a number plus its opposite is 0, and a nonzero number times its reciprocal is 1. (first used: 3. Use the properties of real numbers: why the shortcuts work)
reciprocal — For a nonzero number, the number that multiplies it to give 1, so the reciprocal of 4 is one fourth. (first used: 3. Use the properties of real numbers: why the shortcuts work)
like terms — Terms that have the same variable raised to the same exponent, such as 5x and negative 6x, which the distributive property lets you add. (first used: 3. Use the properties of real numbers: why the shortcuts work)
variable — A letter, such as x, that stands for a number that can change or that is not yet known. (first used: 3. Use the properties of real numbers: why the shortcuts work)
product rule — When two powers with the same base are multiplied, keep the base and add the exponents. (first used: 4. Build the exponent rules on the powers of two)
quotient rule — When a power of a nonzero base is divided by another power of the same base, keep the base and subtract the exponents. (first used: 4. Build the exponent rules on the powers of two)
power rule — When a power is raised to another power, keep the base and multiply the exponents. (first used: 4. Build the exponent rules on the powers of two)
power of a product — When a product is raised to a power, every factor is raised to that power. (first used: 4. Build the exponent rules on the powers of two)
zero exponent — Any nonzero base raised to the exponent 0 equals 1. (first used: 4. Build the exponent rules on the powers of two)
negative exponent — An exponent that calls for a reciprocal of the matching positive power; it never makes the value itself negative. (first used: 4. Build the exponent rules on the powers of two)
algebraic expression — A combination of numbers, variables, and operation signs with no equals sign. (first used: 5. Simplify and evaluate: distribute, combine, then substitute)
evaluate — To find the value of an expression by substituting a number for each variable and then following the order of operations. (first used: 5. Simplify and evaluate: distribute, combine, then substitute)
coefficient — The number that multiplies a variable in a term, such as the 3 in 3x squared, which is applied after any exponent on the variable. (first used: 5. Simplify and evaluate: distribute, combine, then substitute)

Summary

Every natural number is an integer, every integer is rational, and every rational number is real.
The irrational numbers are the real numbers that are not rational, such as the square root of 2 and pi.
Evaluate grouping, then exponents, then multiplication and division left to right, then addition and subtraction left to right.
A minus sign outside the parentheses is applied after the power; a minus sign inside is part of the base.
The commutative, associative, distributive, identity, and inverse properties hold for every real number.
Multiplying powers adds exponents; dividing subtracts them; a power of a power multiplies them.
A zero exponent gives 1, and a negative exponent gives a reciprocal, never a negative number.
Distribute a negative to every term, combine like terms, and check by substitution.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

Sort these five numbers into the smallest family each belongs to: −9, 2⁄3, √25, √7, and 1.5.
Evaluate 20 − 2 · 3², and then evaluate −3² + (−3)². Show each step in order.
Simplify 3⁴ · 3² ⁄ 3³ with the exponent rules, and check your answer by arithmetic. Then write 5⁻² as a fraction.
Simplify 4(2y − 3) − 3(y − 5), and then evaluate your simplified form at y = −2. Check by substituting y = −2 into the original.

Answers (worked reasoning — read after attempting)

The number −9 is the opposite of a counting number, so it is an integer; it is not whole. The fraction 2⁄3 is a quotient of two integers, so it is rational. The root √25 = 5, because 5² = 25, so √25 is a natural number. The root √7 is irrational, because 7 is not a perfect square. Its decimal never stops or repeats. The decimal 1.5 = 15⁄10 = 3⁄2, so 1.5 is rational. The trap was √25: evaluate before you classify.
For 20 − 2 · 3², do the exponent first: 3² = 9. Then multiply: 2 · 9 = 18. Then subtract: 20 − 18 = 2. For −3² + (−3)², the first term is a negated power: −3² = −(3²) = −9. The second term has a negative base: (−3)² = 9. Then add: −9 + 9 = 0. The two terms are opposites, which is exactly the difference the parentheses make.
The product rule gives 3⁴ · 3² = 3⁶, and the quotient rule gives 3⁶ ⁄ 3³ = 3³ = 27. Check by arithmetic: 3⁴ = 81 and 3² = 9, so 81 · 9 = 729, and 729 ⁄ 27 = 27. For the negative exponent, 5⁻² = 1⁄5² = 1⁄25, a small positive number, not −25.
Distribute both factors: 4(2y − 3) = 8y − 12, and −3(y − 5) = −3y + 15, because −3 times −5 is +15. Combine like terms: 8y − 3y = 5y, and −12 + 15 = 3. The simplified form is 5y + 3. At y = −2 it gives 5(−2) + 3 = −10 + 3 = −7. The original gives 4(−4 − 3) − 3(−2 − 5) = 4(−7) − 3(−7) = −28 + 21 = −7. Both give −7, so the simplification checks.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz — and next week these properties become the rule for solving: do the same thing to both sides.

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