UNIT 1 MILESTONE 1 - UNIT 1 — MILESTONE 1

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UNIT 1 — MILESTONE 1

Score21/21 You passed this Milestone 21questions were answered correctly.1

Consider the following set of real numbers:

Which of the following contains all of the irrational numbers in the set? 

RATIONALE

Rational numbers can be expressed as a ratio of two integers, and are characterized by either terminating or repeating decimal patterns, such as 0.375 or 0.3333... Irrational numbers are characterized by a non-terminating, non-repeating decimal pattern.Evaluate each number and determine whether it is rational or irrational.

Irrational: pi has a non-terminating, non-repeating decimal pattern. 

Irrational:  evaluates to -1.7320508... It has a non-terminating, non-repeating decimal pattern. 

Rational: The digits terminate after the 5 in the tenths place.

Rational: The digits terminate after 0.

Rational: This is a ratio of two integers, 1 and 7. 

Irrational: This evaluates to 2.2360679... It has a non-terminating, non-repeating decimal pattern.

Rational: The square root of 9 evaluates to the integer 3. 

Rational: This has a repeating decimal pattern. 

The irrational numbers in the set are .

CONCEPT

Real Number Types 2 Write the following expression as a single number. 

9/19/21, 12:19 AMSophia :: Welcome https://app.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/108047282/14 -64 -7 -10 -61

RATIONALE

For this expression, there is a lot to consider here. Follow the order of operations, and evaluate anything inside parentheses and other grouping symbols first. There are two groups. First, the radical symbol groups (8 × 10 + 1) and a set of parentheses groups . Let's evaluate the radical first.  Underneath the radical, there is multiplication and addition. Multiplication comes before addition in the order of operations, so we multiply 8 by 10 to get 80. Next, evaluate the addition.80 plus 1 is 81. Next, evaluate the other set of parentheses.Inside this parentheses, there is an exponent and subtraction. Exponents comes before subtraction in the order of operations, so we square 3 first, which is equivalent to 9. Evaluate the subtraction next.

• minus 1 is 8. Now that we have taken care of all the parentheses and grouping symbols, we will evaluate the radical next. 

The square root of 81 is 9. Evaluate the multiplication next.

• times 8 is 16. Finally , evaluate the subtraction.
• minus 16 is -7. The expression  simplifies to -7.

CONCEPT

Order of Operations: Exponents and Radicals

3 Find the value of this expression.19 -11 -13 17

RATIONALE

To solve this expression, evaluate the exponent for each term. Start with the first term, .Any number taken to the power of zero equals 1, so  is equal to 1. Evaluate the next term, .When the exponent is 1, the value of the term is the same as its base, so  remains 3. Next, evaluate the term , which is the same as

(-4)(-4).

Negative 4 times negative 4 equals positive 16. The last term,  indicates that -1 is multiplied by itself three times.   is equivalent to -1, because when a negative number is multiplied by itself an odd number of times, the answer remains negative.Finally, evaluate the addition and subtraction from left to right, starting with -1 plus 3.-1 plus 3 is 2. Next, add 2 and 16.

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• plus 16 is 18. Finally, evaluate 18 minus -1.

18 minus -1 is the same as 18 plus 1, or 19. 

CONCEPT

Introduction to Exponents 4 The Elster family drove 9.25 hours on the first day of their road trip.How many minutes is this equivalent to?33,300 minutes 555 minutes 9,256 minutes 154 minutes

RATIONALE

In general, we use conversion factors to convert from one unit to another. A conversion factor is a fraction with equal quantities in the numerator and denominator, but written with different units.  We want to convert hours to minutes. We know how many minutes are in 1 hour. We will use this fact to set up a conversion factor.There are 60 minutes in 1 hour so to convert 9.25 hours into minutes, we will multiply by the fraction  .Notice how the fractions are set up. The units of hours will cancel, leaving only minutes. Finally we can evaluate the multiplication by multiplying across the numerator and denominator.In the numerator, 9.25 times 60 equals 555. 9.25 hours is equivalent to 555 minutes.

CONCEPT

Converting Units 5 Simplify the following radical expression.

RATIONALE

fraction numerator 60 space m i n u t e s over denominator 1 space h o u r end fraction

9/19/21, 12:19 AMSophia :: Welcome https://app.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/108047284/14 To simplify this expression, we can rewrite 45 into products of smaller numbers. There are many ways to do this, but it can help to use a perfect square, since they simplify to integers when we evaluate the square root.45 can be written as 9 times 5. Now we can use the Product Property of Radicals to write the factors as separate radicals.The Product Property allows us to write the radical as the product of individual roots. Finally, we can evaluate the square roots.The square root of 9 is 3. The square root of 5 is already written in its simplest form. The fully simplified radical is .

CONCEPT

Simplifying Radical Expressions 6 Perform the following operations and write the result as a single number.

[‐8 + (4 × 6)] + 9 – (10 ÷ 5)

27 -9 23 -5

RATIONALE

Following the Order of Operations, we must first evaluate the operations inside parentheses and brackets. When there are parentheses inside of brackets, evaluate the innermost operations first. In this case, we must multiply 4 by 6 first. 

• times 6 is 24. Now we can finish this set of parentheses/brackets, and add -8 and 24 next.

-8 plus 24 is 16. There is another set of parentheses to evaluate next: 10 divided by 5.  10 divided by 5 is 2. Now there is only addition and subtraction. Evaluate these as you see them reading left to right. Add 16 and 9 next.16 plus 9 is 25. Finally, subtract 2 from 25.  25 minus 2 is 23. 

CONCEPT

Introduction to Order of Operations 7

Consider the following expression:

What is the value of this expression when x = 2?1 -2 2 -1

RATIONALE