Download Grade 8 Math - Oak Meadow School

Grade 8
Math
Oak Meadow
Coursebook
Oak Meadow, Inc.
Post Office Box 1346
Brattleboro, Vermont 05302-1346
oakmeadow.com
Item #b084010
Grade
8
Contents
Introduction...................................................................... vii
Lesson 1: Using a Calculator........................................... 1
Using a Calculator to Perform Basic Operations With Decimals
Converting Common Fractions To Decimals With a Calculator
Lesson 2: Converting Decimals to Percents................. 13
Finding a Percent of a Number Using a Calculator
Finding Percents in Word Problems
Lesson 3: Squares and Exponents................................. 23
Square Roots
Lesson 4: Negative Numbers.......................................... 33
Directions and the Number Line
Adding and Subtracting Signed Numbers
Adding Signed Numbers
Lesson 5: Multiplying Signed Numbers........................ 43
Dividing Signed Numbers
Lesson 6: Order of Operations...................................... 51
Addition and Subtraction
Parentheses in the Order of Operations
Order of Operations with Multiplication
Lesson 7: Order of Operations with Division............... 63
Order of Operations with Exponents
PEMDAS—The Order of Operations
Fractions in the Order of Operations
iii
Contents
Grade 8 English
Lesson 8: Equations........................................................ 77
Addition Rule of Equations
The Subtraction Rule of Equations
Lesson 9: Division Rule For Equations.......................... 85
The Multiplication Rule For Equations
Lesson 10: Using Two Rules to Solve Equations.......... 93
Variables, Terms, and Coefficients
Lesson 11: Combining Like Terms............................... 103
Negative Coefficients
Two-step Evaluation Problems
Lesson 12: Using a Compass....................................... 113
Drawing an Equilateral Triangle
Constructing a Perpendicular Bisector
Drawing a Geometric Design
Lesson 13: Angles.......................................................... 125
Right Angles
Types of Angles
Measuring Angles
Copying and Bisecting Angles by Construction
Bisecting an Angle
Lesson 14: Ratios.......................................................... 137
Using Ratios to Compare Prices
Proportions
Lesson 15: Using Proportions in Word Problems..... 145
Converting Units in Proportions
Lesson 16: Polygons...................................................... 153
Triangles
Classifying by Angles
Classifying by Sides
Sum of the Angles
Congruent and Similar Triangles
Constructing Triangles
iv
Oak Meadow
Grade 8 English
Contents
Lesson 17: Formulas..................................................... 165
Area of a Rectangle
Distance, Rate, and Time
Transforming Formulas
Lesson 18: First Semester Exam................................... 173
Lesson 19: Creating Equations from
Word Problems......................................................... 181
More Equations From Word Problems
Lesson 20: Equations for Parts of Numbers.............. 193
Word Problems For Parts of Numbers
Lesson 21: Symbols of Inclusion.................................. 201
Fractions in Brackets
Lesson 22: Factors of Whole Numbers....................... 209
Prime Numbers
Third and Fourth Roots of Numbers
Graphing Inequalities
Lesson 23: Evaluating Variables With Exponents...... 219
Multiplying Exponents
Lesson 24: Equations With Exponents....................... 227
Multiple-Term Equations
Combining Terms on Both Sides
Lesson 25: Areas of Triangles....................................... 235
The Pythagorean Theorem
Lesson 26: Pi and the Measures of a Circle................ 245
Pi and the Area of a Circle
Lesson 27: Geometric Solids........................................ 255
Volume of Solids
Lesson 28: Interest and Principal................................ 265
Compound Interest
Lesson 29: Finding the Mean and the Median........... 273
Advanced Ratio Problems
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v
Contents
Grade 8 English
Lesson 30: Fractions with Exponents......................... 283
Roots of Fractions
Lesson 31: Exponents with Negative Bases................ 289
Roots of Negative Numbers
Lesson 32: Scientific Notation..................................... 299
Multiplying with Scientific Notation
Lesson 33: Functions.................................................... 307
Finding Functions
Lesson 34: Recangular Coordinates............................ 321
Graphing a Line for an Equation
Lesson 35: Base 2 Numbers......................................... 333
Writing a Base 2 Number
Lesson 36: Second Semester Exam.............................. 343
Appendix........................................................................ 353
Answer Keys: Skill Practice and Application Practice Problems
B-Tests For Enrolled Students
vi
Oak Meadow
Grade 8
3
Squares and Exponents
When we multiply one number by the same number, we call this a square.
We can find the square of any number by multiplying that number by itself. So the square of 3 is 9, because 3 x 3 = 9, and the square of 5 is 25,
because 5 x 5 = 25.
We call these numbers squares because this is how they are derived. For example, we can show 3 times 3 like this:
Each smaller square is 1 unit, so each side is 3 units long. When we count
all the smaller squares within the figure we find there are 9 squares in all.
So we can say that 3 times 3 equals 9, or the square of 3 is 9.
When we want to indicate repeated multiplication of any number, we do
2
this with an exponent. We indicate the square of 3 by writing 3 . The small
2 that is at the upper right of the 3 is called an exponent, and the 3 is called
the base. The exponent indicates how many times the number is to be used
in the repeated multiplication. If we write this out in a horizontal format,
we can see clearly what this means:
32 = 3 × 3 = 9
We read numbers with exponents in the following way:
We read 32 as “3 to the second power,” or “3 squared.”
We read 53 as “5 to the third power,” or “5 cubed.”
We read 64 as “6 to the fourth power.”
We read 25 as “2 to the fifth power.”
23
Lesson 3
Grade 8 Math
Squares and
Exponents
(continued)
Example 1: What is the value of 43?
43 means 4 × 4 × 4. We can write this out by hand like this:
4
× 4
16
× 4
64
So we can say that 43 = 64. We can do this on a calculator as follows:
Step 1: Clear the calculator to make sure no previous numbers are
entered.
Step 2: Enter 4.
Step 3: Press the × key to indicate multiplication.
Step 4: Enter 4.
Step 5: Press the × key to indicate multiplication.
Step 6: Enter 4.
Step 7: Press the = key to display the answer, which is 64.
We can also use exponents with decimal fractions, as follows:
Example 2: What is the value of 3.22?
Step 1: Clear the calculator to make sure no previous numbers are
entered.
Step 2: Enter 3.2.
Step 3: Press the × key to indicate multiplication.
Step 4: Enter 3.2.
Step 5: Press the = key to display the answer, which is 10.24.
If the base is larger, even a small exponent can cause the value of a number
to increase very quickly, as in the following example:
Example 3: What is the value of 564?
If we look at the sequence of this at each stage of multiplication, we can
see how quickly the size of the number increases:
24
Oak Meadow
Grade 8 Math
Lesson 3
56 × 56 = 3,136
3,136 × 56 = 175,616
175,616 × 56 = 9,834,496
Squares and
Exponents
(continued)
When a value increases quickly by means of an exponent, we say that the
increase in the value is exponential, or that the value is increasing exponentially.
Square Roots
When we square a number, we multiply it by itself. To find a square root of
a number, we find a number which, when multiplied by itself, equals the
original number. Thus, the square root of 9 is 3, because 3 x 3 = 9. To
indicate the square root of a number, we use a square root symbol, which
looks like this:
Using this symbol, we read 2 5 as “the square root of 25.” We can write
the solution to the problem like this:
25 = 5
This means that when we multiply 5 times itself, we get 25.
Example 1: What is the square root of 49?
To solve this, we have to ask ourselves, “What number multiplied by itself
equals 49?” We know that 7 x 7 = 49, so we can say that the square root
of 49 is 7.
Example 2: What is the value of
64 ?
Since 8 x 8 = 64, the square root of 64 is 8.
Finding square roots for familiar numbers in the multiplication table is not
difficult, but when we encounter other numbers we can use the square
root key on a calculator. Look at the following example:
Example 3: What is the value of
729 ?
Step 1: Clear the calculator to make sure no previous numbers are
entered.
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25
Lesson 3
Grade 8 Math
Squares and
Exponents
(continued)
Step 2: Enter 729.
Step 3: Press the
key to indicate the square root.
Step 4: The calculator displays the square root, which is 27.
You can also use your calculator to find the square root of a decimal
number, as follows:
Example 4: What is the value of
32.64 ?
Step 1: Clear the calculator to make sure no previous numbers are
entered.
Step 2: Enter 32.64.
Step 3: Press the
key.
Step 4: The calculator displays the square root, which is 5.7131427.
Since square roots of decimal fractions will often be large
decimal fractions themselves, we usually round them off to
two decimal places. Using the basic rule of rounding, we get
5.71.
26
Oak Meadow
Grade 8 Math
Lesson 3
Skill Practice A
Use your calculator to determine the value of the following terms. Do not round off decimal
answers.
1. What is the value of 152?
2.What is the value of 483?
3.What is the value of 6.34?
4.What is the value of 105?
5.What is the value of 844?
6.What is the value of 2.73?
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Lesson 3
Grade 8 Math
Skill Practice A continued
7. What is the value of 164?
8. What is the value of 0.23?
9. What is the value of 754?
10.What is the value of 193?
11. What is the value of 1.74?
12. What is the value of 162?
28
Oak Meadow
Grade 8 Math
Lesson 3
Skill Practice B
Use your calculator to determine the value of the following terms. Round off decimal fractions to
two decimal places.
1.
What is the square root of 169?
2.
What is the value of
3.
What is the the value of
4.
What is the square root of 14.44?
5.
What is the value of
6.
What is the square root of 10,000?
Oak Meadow
640 ?
1,0 2 4 ?
196 ?
29
Lesson 3
Grade 8 Math
Skill Practice B continued
7.
What is the square root of 34.81?
8.
What is the value of
360 ?
9.
What is the value of
92.16 ?
10.
What is the square root of 625?
11.
What is the value of
12.
What is the square root of 15.21?
30
256 ?
Oak Meadow
Math 8
Lesson 3 - 11 Lesson 3
Grade 8 Math
Lesson 3 Review
REVIEW 2
Use a calculator to solve decimal problems, and solve common fraction problems by hand.
Use
calculator
totosolve
decimalplaces.
problems, and solve common fraction
Round
offalonger
answers
two decimal
problems by hand. Round off longer answers to two decimal places.
1. 62.4 − 18.29
2. 1 2 . 4 5
3. 3 37 ÷ 4 57
4. 0.63 + 3.49
5. 2 31 + 5 34
6. 7.9 − 0.65
7. 9.76 × 14
8. .7 32.1
9. 7 × 2 51
10. 42 ÷ 3.6
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× 9.2
11. 7 51 − 4 54
12. 12.4 + 7.7
31
Lesson 3
Grade 8 Math
Math 8
Lesson 3 - 12
Use your
calculator
to solve the following problems.
Lesson
3 Review
continued
Use your calculator to solve the following problems.
13.
576
15. 15 4
14. 8 . 7 3
16.
475.24
17. 25% of the people at Becky’s church are under 18 years old. If there are 320 people in the
17. church,
25% ofhow
the many
people
Becky's
areatunder
18? church are under 18 years old. If there are
320 people in the church, how many are under 18?
18.Dennis
to cook,
and he
recipes
in his collection.
8 of his recipes
arehis
for cookies.
18.
Dennisloves
loves
to cook,
andhashe60has
60 recipes
in his collection.
8 of
What
percent
of his
total collection
are cookieofrecipes?
Round
off the answer
a whole number.
recipes
are for
cookies.
What percent
his total
collection
are tocookie
recipes? Round off the answer to a whole number.
19. Ricky bought a pair of pants for $29.95, and the clerk added sales tax of 6%. If he gave the
19. clerk
Ricky
bought
a pairchange
of pants
forhe$29.95,
and the(Round
clerkoffadded
sales
$40,
how much
should
have received?
the sales
tax totax
theof
nearest cent)
6%. If he gave the clerk $40, how much change should he have
received? (Round off the sales tax to the nearest cent)
20.Craig’s cat Amadeus gave birth to 7 kittens. 5 of the kittens are female and 2 are male.
What percent of Amadeus’s kittens are female? (Round off your answer to the nearest one percent)
20. Craig's cat Amadeus gave birth to 7 kittens. 5 of the kittens are female
and 2 are male. What percent of Amadeus's kittens are female? (Round
off your answer to the nearest one percent)
32
Oak Meadow
Grade 8
4
Negative Numbers
We can show a sequence of numbers using a number line like the one below. The arrows on each end of the number line show that the line continues without end in both directions. The numbers that are marked on the
number line are called whole numbers, and they include the number zero.
The whole numbers to the right of zero on the number line are known as
positive numbers. To the left of zero are negative numbers. We can show them
on the number line as follows:
Negative numbers are all the numbers less than zero, and they always have
a minus sign in front of them. Each positive number has an opposite negative number. The opposite of 5 is -5, and the opposite of 38 is - 38. Zero
is between the positive and the negative numbers, so it is neither positive
nor negative. All of the whole numbers on the number line (the positive
whole numbers, the negative whole numbers, and zero) are called integers.
Between the integers are all of the fractional numbers, both common and
decimal fractions. We can call all of these numbers (except zero) signed
numbers, because they have a sign, either positive or negative.
Positive numbers can be written with or without a plus sign. If there is no
sign before a number, then it is a positive number.
33
Lesson 4
Grade 8 Math
Negative
Numbers
(continued)
Directions and the Number Line
When we move to the right on the number line, we are moving in a positive
direction. When we move to the left on the number line, we’re moving in
a negative direction. A positive direction is indicated by a plus sign, and a
negative direction is indicated by a minus sign. For example, if we start
with a +2 and move 3 units in a negative direction, the operation looks like
this:
We could write this as 2 - 3, and we would say that 2 - 3 = -1.
If we start at +2 and move 3 units in a positive direction, the operation
looks like this:
We could write this as 2 + 3, and we would say that 2 + 3 = 5.
Adding and Subtracting Signed Numbers
As we mentioned in the previous section, signed numbers include all positive
and negative numbers. When we add or subtract signed numbers, we not
only have to consider the operation itself (whether it’s addition or subtraction), but we also have to consider the signs of the numbers themselves.
To complete these operations correctly, mathematicians have developed
certain rules and definitions. Let’s first look at a definition:
Absolute Value—The value of an integer without its sign.
The absolute value of -3 and +3 is 3
The absolute value of -12 and +12 is 12
Now that we know the meaning of absolute value, let’s look at the rules and
some examples of adding signed numbers.
34
Oak Meadow
Grade 8 Math
Lesson 4
Adding Signed Numbers
Rule 1: If two numbers have the same sign, add their absolute values and
give the sum the sign of the original numbers.
Example 1: 5
Negative
Numbers
(continued)
+3=8
We add the absolute value of 5 and 3 and get 8. Since neither number has
a sign, that means they are both positive, so the answer is also positive.
(We could add the sign and make it +8, but since the original numbers
didn’t have a sign, we leave it off the answer also).
Example 2: -5
+ (-3) = -8
We add the absolute value of 5 and 3 to get 8. Since both signs are negative, we give the answer a negative sign. We put the -3 in parentheses to
show the difference between the addition sign and the sign of the number.
Rule 2: If two numbers have different signs, subtract the smaller absolute
value from the larger and give the result the sign of the number with the
larger absolute value.
Example 3: -4
+ 2 = -2
The first number is negative and the second has no sign, so it’s positive.
We subtract the absolute value of the smaller number from the absolute
value of the larger number to get 2. Since the sign of the larger number is
negative, then we make the answer negative also.
Example 4: 4
+ (-2) = +2
The first number has no sign, so it’s positive, while the second number is
negative. We subtract the absolute value of 2 from 4 and get 2. The larger
number is positive, so the answer is positive.
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35
Lesson 4
Grade 8 Math
Negative
Numbers
(continued)
Subtracting Signed Numbers
To subtract signed numbers, we must remember that subtraction is the
inverse operation to addition. This means that we can use signed numbers
to make any subtraction problem into an addition problem. We can use a
number line to demonstrate this principle:
Example 1: 1
-3
When we subtract 3 from 1, this gives the same result as if we started at 1
and added -3, because we are still moving from 1 in a negative direction on
the number line. So we can say that 1 - 3 = 1 + (-3), and the answer is -2.
Using this principle, there is only one rule for subtracting signed numbers:
Rule 1: To subtract signed numbers, add the first number and the opposite of the number being subtracted.
Example 2: 5
-2
We can use the subtraction rule to add 5 and the opposite of 2, which is -2.
(Since 2 doesn’t have a sign, it’s positive, so the opposite of positive 2 is
negative 2).
5 - 2 = 5 + (-2)
Then we can solve it using the same rules for adding signed numbers that
we have just practiced:
Example 3: 2
-7
5 - 2 = 5 + (-2) = 3
Once again, the 7 doesn’t have a sign, so when we rewrite it as an addition
problem we change the 7 to its opposite, which is -7.
2 - 7 = 2 + (-7)
Then we solve it, using the rule of addition:
2 - 7 = 2 + (-7) = -5
36
Oak Meadow
Grade 8 Math
Lesson 4
Example 4: -8
Example 5: -5
-3
- (-6)
-8 - 3 = -8 + (-3) = -11
Negative
Numbers
(continued)
The sign of the second number is given, so we change the problem to
addition and reverse the sign of the 6, as follows:
Example 6: 8
- (-4)
-5 - (-6) = -5 + (+6) = +1
8 - (-4) = 8 + 4 = 12
For Enrolled Students
Before you continue with Lesson 5, send the Review and Test for lessons 3
and 4 to your teacher.
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37
Lesson 4
Grade 8 Math
Notes
38
Oak Meadow
Grade 8 Math
Lesson 4
Skill Practice A
Solve the following expressions.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10+(−4)=
−15+12=
−11+3=
8+(−8)=
−9+(−3)=
12+4=
14+(−6)=
−7+(−12)=
5+(−6)=
10. −4+(−3)=
11. 7+(−2)=
12. 3+(−4)=
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39
Lesson 4
Grade 8 Math
Skill Practice B
Solve the following expressions.
1.
2.
3.
4.
5.
6.
7.
8.
9.
−2−(+3)=
3−(+9)=
−11−(−4)=
12−(+8)=
15−(+7)=
−4−(−9)=
10−(+10)=
8−(+2)=
3−(+6)=
10. −7−(−3)=
11. 9−(−2)=
12. 5−(+6)=
40
Oak Meadow
Math 8
Lesson 4 - 9
Grade 8 Math
Lesson 4
TEST 2
Lesson 4 Test
Use a calculator to solve decimal problems, and solve common fraction
problemstobysolve
hand.
Round
off longer
answers
to two
decimal
places.
Use a calculator
decimal
problems,
and solve
common
fraction
problems
by hand.
Round off longer answers to two decimal places.
1. − 2 + ( − 5 )
2. 13.6 + 8.9
3. 14 − ( + 9 )
4. − 3 − ( − 7 )
5. 5 ÷ 2 51
6. .409 × .8
7. 2 31 × 21
8. 0.3 24.9
9. 4 + ( − 6 )
10. 14 ÷ 2.3
11. 8.1− 0.42
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12. 2 8 . 7 3
× 5.4
41
Lesson 4
Grade 8 Math
Lesson 4 Test
13.Write
2 35
as a decimal.
2
14.Write 3 as a decimal rounded off to hundredths
15. What is 38% of 1,200?
16.What percent is 47 out of 50?
42
Oak Meadow
Grade 8
23
Evaluating Variables
with Exponents
We have learned how to evaluate an algebraic expression by substituting
numerical values for the variables, as in the following example:
Example 1: Evaluate 3x if x = 2
We substitute the given value into the expression and simplify:
3x=3(2)=6
If one of the variables is the base of an exponential expression, we can
evaluate it the same way, by substituting the given value for the variable.
Example 2: Evaluate x2 if x = 3
We substitute the given value into the expression and simplify:
x2=32=3⋅3=9
This same process applies if the variable is the exponent itself, rather than
the base, as in the following example.
Example 3: Evaluate 5x if x = 2
We substitute the given value into the expression and simplify:
5x=52=5∙5=25
This same process also applies when we are evaluating roots. Look at the
following examples:
Example 4:
Evaluate
4
n if n = 81
We substitute the given value into the expression and simplify:
4
n = 4 81 = 3
219
Lesson 23
Grade 8 Math
Evaluating
Variables with
Exponents
We can also substitute for the power of the root, as follows:
Example 5: Evaluate
n
1 2 5 if n = 3
We substitute the given value into the expression and simplify:
(continued)
n
125 = 3 125 = 5
Multiplying Exponents
We have learned that an exponential expression indicates a process of
4
multiplication. For example, the expression 3 means 3∙3∙3∙3, and the
product is 81.
If we multiply two exponential expressions, one solution is to simplify each
term and add the results, like this:
22∙24=(2∙2)(2∙2∙2∙2)=(4)(16)=64
Notice that in this process, the base (2) is multiplied by itself 6 times. This
is the same value as the total of the exponents of the two original expressions (2 + 4 = 6). This leads us to a rule for multiplying exponents:
To multiply exponential terms with the same base, add the exponents.
Example 1: 42∙43
Since both expressions have the same base (4), we add the exponents.
2 + 3 = 5, so the result is 45 .
Example 2: a4∙a6
Since variables take the place of numbers, we can apply the same rule with
variables that have exponents. Both expressions have the same base, so
we add the exponents. 4 + 6 = 10, so
Example 3: 34∙46
a4∙a6=a10
The rule for multiplying exponents only applies when both terms have the
same base. These terms have different bases. One base is 3 and the other is
4. We can’t add the exponents, so we leave it as is: 34∙46.
Example 4: a4∙b6
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Oak Meadow
Grade 8 Math
Lesson 23
Although both terms contain variables, the variables are different, so the
terms have different bases. We can’t add the exponents, so we put the
two terms together as a4b6, to indicate multiplication.
Example 5: x2∙x
Evaluating
Variables with
Exponents
(continued)
The bases are the same, so we can add the exponents. The first term has
an exponent of 2. The second term doesn’t have an exponent. When a variable doesn’t have an exponent, we treat it as if it has an exponent of 1.
When we add the exponents, 2 + 1 = 3, so the value is x .
3
If the multiplication of two numbers results in a very large exponent, we
generally don’t simplify it completely; we just leave it in exponential form.
To multiply it would result in a very large number that would be more subject to error than just leaving it as the base and the exponent alone. For
example, if we are mulitplying 812∙815, we leave the answer as 827.
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221
Lesson 23
Grade 8 Math
Notes
222
Oak Meadow
Grade 8 Math
Lesson 23
Skill Practice A
Evaluate the following expressions for the given values:
1.
a2 if a = 4
2.
6 x if x = 3
3.
3
d if d = 343
4.
n
2 5 6 if n = 4
5.
p3 if p = 8
6.
3 w if w = 4
7.
4
c if c = 1,296
8.
e
7 2 9 if e = 3
9.
g4 if g = 8
10. 1 8
m
if m = 3
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223
Lesson 23
Grade 8 Math
Skill Practice B
Solve the following using the rule for multilication of exponents. In this practice, you only need to
indicate the correct base and exponent. For example, the correct answer for 44∙43 would be 47.
1.
23∙25
2.
b5∙b9
3.
52∙63
4.c2∙d2
5.
y3∙y
6. 47∙48
7.
e3∙e9
8.
92∙75
9.
x12∙y10
10. g3∙g4∙g
224
Oak Meadow
Grade 8 Math
Lesson 23
Lesson 23 Review
1. Is 31 a prime number?
2.4 times a number is 17 less than 29. What is the number?
3.If
3 x + 8 = 2 , what is the value of 21 x + 1 2 ?
4.If an airplane is flying at the rate of 25 miles in 3 minutes, how long will it take it to travel
1,500 miles?
5.What is the root of
3
1,3 3 1 ?
6.Simplify: 2[8(5)+5(7−3)]+3[(6−3)+2(4)]
7.Evaluate 12x if x = 3
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225
Lesson 23
Grade 8 Math
Lesson 23 Review continued
8.Evaluate 3b−2bc+13 for b = 8 and c = 3.
9. The sum of twice a number and 16 is 84. What is the number?
10.The ratio of nuts to bolts in the box is 3:2. If there are 60 nuts in the box, how many bolts
are there?
11. a4∙a6
1
12. 8
c=−4
1
13. 2 3
2
z+18=6
45
14. 5 = w
15. 512∙518
3
16. 8
226
+6d=−2 14
Oak Meadow
Grade 8
24
Equations with
Exponents
We have learned how to solve equations with single variables, such as the
following:
x+2=83x=124x+3=11
In all of these equations, the variable does not have an exponent. In this
lesson, we will explore equations in which the variable has an exponent.
Look at the following example:
Example 1: Solve this equation for x: x2=9
In this equation, we know the value of x2, but we need to find the value
of x. To reduce x2 to x and solve the equation, we have to use the basic
rule of equations: whatever we do to one side of the equation, we must do to the
other side. To reduce x2 to x, we take the square root of x2, which is x. So
if we take the square root of x2, then we have to take the square root of
9, which is 3. This seems simple enough, but when we consider negative
numbers, there are actually two solutions to the problem:
√9=3 because (3)(3)=9
√9=−3 because (−3)(−3)=9
For this reason, when we solve an equation that contains an exponent
that is a square, the answer is given as “plus or minus” the value. This is
written as follows:
x2 =9
x =±3
The answer is read as “plus or minus 3.”
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Lesson 24
Grade 8 Math
Equations with
Exponents
(continued)
Example 2:
Solve this equation for y: y2=119
The square root of 119 is not an even value. Round off the answer to two
decimal places and put the ± symbol in front of it, as follows:
y2=119
y=±10.91
Example 3: Solve this equation for d: d2+6=22
First, we use the subtraction rule for equations:
d2+ 6 = 22
− 6 = −6
d2= 16
Then we take the square root of both sides:
Multiple-Term Equations
d2=16
d=±4
When an equation has more than one term on either side, we must first
combine the like terms and then solve it using one or more of the rules for
equations.
Example 1: 3x−4−x+6=42
Step 1:We combine all the like terms:
3x−4−x+6=4
2x+2=42
Step 2:We use the necessary rules for equations to separate the variable
and find the answer:
2x + 2=42
− 2=−2
2x=40
y=6
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Grade 8 Math
Lesson 24
Example 2: 2y−8+y+17=27
2y−8+y+17=27
3y+9=27
−9=−9
3y=18
y=6
Equations with
Exponents
(continued)
Combining Terms on Both Sides
To solve some equations, you have to combine like terms on both sides, as
in the following examples:
Example 1: 4x−2+x+6=3x+6
In these kinds of equations, the object is to get the variables on one side
of the equation and the numbers on the other side. If we follow a consistent process, the equation is not difficult to solve:
Step 1: Combine the like terms on both sides.
4x−2+x+6=3x+6
5x+4=3x+6
Step 2:Using rules for equations, move the variables to one side of the
equation.
5 x +4=3 x +6
−3 x
=− 3 x
2 x +4=6
Step 3: Using rules for equations, move the numbers to the other side of
the equation and solve
2 x +4=6
− 4 =− 4
2 x =2
x =1
For Enrolled Students
Before you continue with Lesson 25, send the Review and Test for lessons
23 and 24 to your teacher.
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Lesson 24
Grade 8 Math
Skill Practice A
Solve the following equations for the variable without the exponent.
2
= 25
2
= 106
3. y
2
+ 6 = 42
4. b
2
+ 9 = 47
1. x
2. d
5. p2−14=43
6. a2−12=37
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Lesson 24
Skill Practice B
1. 4b+6−14+2b=46
2. 7x−9−3x+5=28
3. 18−2a−12+9a=27
4. 5g−2g+15−9=45
5. 2c+6−c+8=42
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Lesson 24
Grade 8 Math
Skill Practice C
1. 2x+6=x+4
2. −3y−2+7y+7=2y+23
3. 4b+6−2b=23+3b−7
4. −d+2+3d−7=18−d+7
5. 2w+6=−5w−16+w+20+4w
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Lesson 24
Lesson 24 Test
Use a calculator to solve decimal problems. Round off longer answers to two decimal places.
Reduce common fractions to lowest terms.
1
1. 2e+ 4
=−3 12
2. x2+3=19
3. 39∙313
4. 5[2(3)+4(6−1)]−6(17−13)
3
5. 8
27
=a
6.Evaluate 9x if x = 4
1
7. 1 2
d−5=10
8.5 times a number is 25 less than 110. What is the number?
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Lesson 24
Grade 8 Math
Lesson 24 Test continued
9. What is the root of
3
2,1 9 7 ?
10. 5w+7−w=14+3w−8
1
11.If 2x+ 2
1
12. 6
=4, what is the value of 4x−8?
p=−2 13
13. Is 43 a prime number?
14.Evaluate a2−3a+6 for a = 4.
15. The sum of twice a number and 19 is 47. What is the number?
16. n3∙n5
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