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UNIT 1 — MILESTONE 1
What is the following expression equivalent to?
RATIONALE
First, apply the Power of a Power Property of Exponents, which states that when an exponent is
raised to another exponent, you can multiply the exponents. Therefore, multiply
evaluate the first part,
is equivalent to
times
to
.
. Next, combine the two terms. The Product Property of Exponents states
that if two expressions with the same base are multiplied together, you can add the exponents. Add
the exponents
plus
is
and
to evaluate the two terms.
, which becomes the final exponent.
CONCEPT
Properties of Exponents
2
Simplify the following radical expression.
RATIONALE
To simplify this expression, we can use the Product Property of Radicals to separate the
expression into two radicals.
The cube root of
can be written as the cube root of
times the cube root of
. Next, we
can write each radical expression using a fractional exponent in order to simplify. The index of
the radical determines the denominator of the fractional exponent. The index here is , so each
expression underneath the radical will be raised to the
power.
Now that we have changed our original expression from a radical to fractional exponents, we
can evaluate and simplify the two expressions that are raised to the
to the power of
simplify
times
power.
evaluates to 3 because 3 raised to the 3rd power is
(
). To
, we multiply the two exponents together.
equals
and
is simply . The expression
simplifies to
.
CONCEPT
Applying the Properties of Radicals
3
Write the following expression as a single integer.
RATIONALE
To solve this problem, we can use Order of Operations. First consider the sign of the first
number. There is a negative number outside the parentheses, which means we must take the
opposite of the number inside the parentheses.
The opposite of negative
is positive , so
evaluates to . Next, subtract
from
.
minus
equals
.
CONCEPT
Adding and Subtracting Positive and Negative Numbers
4
Simplify the following expression in terms of fractional exponents and write it in the form
.
RATIONALE
Radical expressions can be rewritten using fractional exponents by corresponding the index of
the radical to the denominator of the fractional exponent. Begin by writing the expression
underneath the radical. This entire expression will be raised to a fractional exponent power.
The index of the radical is 3. In other words, this is the third-root. This means that the entire
expression will be raised to the power of
the outside exponent of
Once the exponent of
exponents
In this case,
and
. Next, to write this in the form
, distribute
to the powers of 10 and .
is distributed to both terms, we can simplify by multiplying the
for the term of
becomes
.
. The final expression is
.
CONCEPT
Fractional Exponents and Radicals
5
Julie bought the ingredients to make chicken soup, and wanted to make a triple batch, which would be 24
cups of soup. A quick Google search told her that this was 246.5 cubic inches. She hoped the soup pot below
would be big enough.
The soup pot is 8.5 inches tall with a radius of 4 inches.
What is the volume of the soup pot? Answer choices are rounded to the nearest tenth cubic inch.
907.9 cubic inches
427.3 cubic inches
106.8 cubic inches
213.6 cubic inches
RATIONALE
Recall that the volume of the pot can be represented with the formula for the volume of a
cylinder, where is the value, is the radius of the base, and is the height. First,
substitute
for and
for .
Once we have the given values plugged into the appropriate places, we can evaluate the
formula. Following Order of Operations, we will first square the radius of .
squared equals . Next, we can multiply the remaining values. Using a calculator with a
pi button ( ) will be the most accurate; otherwise we can use the value
.
Multiplying times
times
is approximately
. Rounded to the nearest tenth
cubic inch, the volume of the soup pot is 427.3 cubic inches.
CONCEPT
Volume
6
Evaluate the following expression.
RATIONALE
Evaluate each absolute value first. Absolute value is always non-negative. Rewrite the number without any
negative sign if there was one.
The absolute value of -3 is 3. The absolute value of -1 is 1. Now we can add these two numbers
3 plus 1 is 4. Next, multiply 2 by 4.
2 times 4 is 8.
CONCEPT
Introduction to Absolute Value
7
Write the expression as a single power of b.
RATIONALE
Start by simplifying the terms in the parentheses. Using the Quotient Property of Exponents, divide the two
terms that have the same base by subtracting the exponents, and .
When subtracting fractions with a common denominator, subtract across the numerators and leave the
denominator the same.
minus equals . Next, apply the Power of Property of Exponents to multiply the two exponents and write as
a single power.
times equals . Lastly, rewrite the fraction in its simplest form.
simplifies to , so the expression simplifies to .
CONCEPT
Properties of Fractional and Negative Exponents
8
How can the following expression be simplified and written without negative exponents?
RATIONALE
By completing a series of steps, this expression can be written so that no negative exponents are included.
The first step is to write the numerator as a single power of . You can combine the two terms by using the
Product Property of Exponents, which states that if two expressions with the same base are multiplied
together, you can add the exponents. Therefore, add the exponents and to evaluate the two multiplied
terms.
plus equals , which becomes the new exponent in the numerator. Next, divide the numerator by the
denominator and write this as a single power of . To do this, use the Quotient Property of Exponents, which
says that when you divide two expressions with the same base, you can subtract the exponents. Therefore,
evaluate minus .
Be careful when subtracting negative numbers! minus can be thought of as plus .
plus is equal to . This is the new exponent for . Lastly, write this without any negative exponents.
Write the expression as a fraction with in the numerator, and change the sign of the exponent from negative
to positive. This is the simplified expression without any negative exponents.
CONCEPT
Negative Exponents
9
Perform the following operations and write the result as a single number.
RATIONALE
Following the Order of Operations, we must first evaluate everything in parentheses and grouping symbols.
When there are brackets or braces, evaluate the innermost operations first. Here, we must evaluate 5 minus
3 first.
5 minus 3 is 2. There are still operations inside grouping symbols to evaluate. Multiplication comes before
addition, so we must evaluate 8 times 2 next.
8 times 2 is 16. Next, we add 4 and 16 to complete the operations inside parentheses.
4 plus 16 is 20. Now there is just division and subtraction. Division comes before subtraction in the Order
of Operations, so we divide 20 by 5 next.
20 divided by 5 is 4. Lastly, add 4 and 6.
4 plus 6 is 10.
CONCEPT
Introduction to Order of Operations
10
The diameter of a hydrogen atom is about meters. A protein molecule has an overall length of 3000 times (or
times) the diameter of a hydrogen atom.
What is the length of the protein molecule, in meters, if it were written in scientific notation?
meters
meters
meters
meters
RATIONALE
To find the length of the protein molecule, multiply by 3000, which can be expressed as .
When multiplying numbers in scientific notation, you must deal with the numbers and 10s separately. First,
multiply and .
times equals . Now you can use the Product Property of Exponents on the 10s and add the exponents.
plus equals , which is the exponent for the base 10. Finally, combine the number part and the power of 10
together.
The length of the protein molecule (in meters), when written in scientific notation, is .
CONCEPT
Multiplication and Division in Scientific Notation
11
Asher owns a square plot of land. He knows that the area of the plot is between 4200 and 4300 square
meters.
Which of the following is a possible value for the side length of the plot of land?
55 meters
45 meters
65 meters
75 meters
RATIONALE
Asher owns a plot of land that is perfectly square. The area of the plot is between 4200 and 4300 square
meters. One way to solve this problem is to find the area for each side length given. The area of a square is a
special case in which the length and the width are the same, so you can find the area by squaring the length.
If the side length of the square is 45 meters, then the area equals 2025 square meters. This is NOT a
possible length for Asher’s plot because 2025 is not between 4200 and 4300.
If the side length of the square is 55 meters, then the area equals 3025 square meters. This is NOT a
possible length for Asher’s plot because 3025 is not between 4200 and 4300.
If the side length of the square is 75 meters, then the area equals 5625 square meters. This is NOT a
possible length for Asher’s plot because 5625 is not between 4200 and 4300.
If the side length of the square is 65 meters, then the area equals 4225 square meters. This is a possible
length for Asher’s plot because 4225 is between 4200 and 4300.
CONCEPT
Area
12
Perform the indicated operations and write your result as a single number.
RATIONALE
For this expression, there is a lot of consider here. Follow the order of operations, and evaluate anything
inside parentheses first. There are two groups. First, a set of parentheses groups , and the radical symbol
groups . Let's evaluate the set of parentheses first.
minus is . Next, evaluate the radical.
Underneath the radical there is multiplication and addition. Multiplication comes before addition in the
order of operations, so we multiply by to get . Next, evaluate the addition.
plus is . Now that we have evaluated all grouping symbols, we will evaluate exponents and radicals next.
Start by squaring .
squared, or , is equal to . Evaluate the radical next.
The square root of is . Now we evaluate the multiplication.
times is . Finally, evaluate the subtraction.
minus is . The expression simplifies to .
CONCEPT
Order of Operations: Exponents and Radicals
13
Which of the following equations is correctly calculated?
RATIONALE
This is the correct answer. The product of two negative numbers is always a positive number. Negative
times negative equals positive .
This is incorrect. The product of a positive and negative number is always negative. The correct product is .
This is incorrect. The quotient of a positive and negative number is always negative. The correct quotient is
.
This is incorrect. The quotient of two negative numbers is always positive. The correct quotient is .
CONCEPT
Multiplying and Dividing Positive and Negative Numbers
14
A football field is a rectangle 80 meters wide and 110 meters long. Coach Trevor asks his players to run from
one corner to the other corner by running diagonally across the field.
What is the distance from one corner of the field to the other corner? Answer choices are rounded to
the nearest meter.
185 meters
190 meters
136 meters
115 meters
RATIONALE
We can use the Pythagorean Theorem to calculate the length of a diagonal. The variables and represent the
sides of the field, and represents the diagonal. First, substitute for and for . (Note that we could also
substitute for and for ).
Once we have the given values plugged into the Pythagorean Theorem, we can evaluate the exponents.
squared is , and squared is . Now we can add these values together.
plus is equal to . Finally, we can take the square root of both sides to find the value of .
When we have a squared term, such as , taking the square root of both sides will cancel this operation.
The square root of is approximately . The distance from one corner of the field to the other, rounded to the
nearest meter, is 136 meters.
CONCEPT
Calculating Diagonals
15
Simplify the following radical expression.
RATIONALE
To simplify this expression, we can rewrite 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.
can be written as times . 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
16
What is the value of the following expression?
RATIONALE
When evaluating higher-order roots, it helps to break down the number underneath the radical into prime
factors.
can be written as Notice that there are five factors of , and they are all underneath a fifth root. This means
the expression underneath the radical simplifies to .
can be simplified to . Lastly, we apply the negative sign in front of the radical.
The original expression can be simplified to .
CONCEPT
Evaluating Radicals
17
The current population of the Earth is estimated to be about 7,440,000,000 people.
Express this number using scientific notation.
RATIONALE
To express this number in scientific notation, you must find two parts: a number between one and ten (it can
be equal to one, but not ten), and a power of . To find the first part, place the decimal after the first non-zero
number, .
The decimal will be placed after the . We will also include the numbers right after the , which were . These
will go immediately after the decimal. To find the second part, you must find the exponent to which will be
raised. To do this, count how many times the decimal is moved from to obtain .
If we start with the decimal right after the , it would take us nine moves to the right to obtain the original
number. Because we must move the decimal point nine times, the exponent is .
Combine the two parts to state the solution in scientific notation. The population of earth is estimated to be
about people.
CONCEPT
Writing Numbers in Scientific Notation
18
What is the value of the following expression?
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 , so is equal to . Evaluate the next term, .
When the exponent is , the value of the term is the same as its base, so to the first power is . Next, evaluate
the term , which is the same as .
Negative times negative equals positive . The last term, indicates that is multiplied by itself three times.
equals , because when a negative number is multiplied by itself an odd number of times, the answer
remains negative. Finally, add all of the terms.
plus plus is . The last step is to subtract .
minus is .
CONCEPT
Introduction to Exponents
19
The Elster family drove 9.25 hours on the first day of their road trip.
How many minutes is this equivalent to?
154 minutes
555 minutes
33,300 minutes
9,256 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 minutes in hour so to convert 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, times equals . 9.25 hours is equivalent to 555 minutes.
CONCEPT
Converting Units
20
Consider the following expression:
What is the value of this expression when ?
RATIONALE
To find the value of this expression when , begin by substituting for every instance of in the expression.
Once all instances of have been substituted with , we can evaluate the expression. The division bar acts as a
grouping symbol, separating the expression in the numerator, , from the expression in the denominator, .
Evaluate them separately before dividing, starting with evaluating the absolute value of .
Recall that the absolute value of a number is the non-negative value. The absolute value of is . Next, divide
the numerator by the denominator.
divided by is , but do not forget about the negative sign in front.
CONCEPT
Operations as Grouping Symbols
21
Consider the following set of real numbers:
Which of the following lists all of the rational 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 nonterminating, non-repeating decimal pattern. Evaluate each number and determine whether it is rational or
irrational.
Rational: The square root of 4 evaluates to the integer 2.
Irrational: evaluates to and has a non-terminating, non-repeating decimal pattern.
Rational: This is a ratio of two integers, and .
Rational: This is an integer.
Rational: This is a ratio of two integers, and .
Rational: This simplifies to the fraction which is a ratio of two integers, and .
Rational: This has a repeating decimal pattern.
Irrational. evaluates to and has a non-terminating, non-repeating decimal pattern.
The rational numbers in the set are
CONCEPT
Real Number Types
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