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Math Problems
Grade 8
Divide numbers written in scientific notation
Which of the following options have the same value as
62
%
62\%
62%
of
45
45
45
? Choose
2
2
2
answers:
\newline
Choose
2
2
2
answers:
\newline
(Choice A)
0.62
×
45
0.62 \times 45
0.62
×
45
A
0.62
×
45
0.62 \times 45
0.62
×
45
\newline
(Choice B)
45
0.62
\frac{45}{0.62}
0.62
45
B
45
0.62
\frac{45}{0.62}
0.62
45
\newline
(Choice C)
(
62
100
)
×
45
\left(\frac{62}{100}\right) \times 45
(
100
62
)
×
45
C
(
62
100
)
×
45
\left(\frac{62}{100}\right) \times 45
(
100
62
)
×
45
\newline
(Choice D)
62
45
\frac{62}{45}
45
62
D
62
45
\frac{62}{45}
45
62
\newline
(Choice E)
62
×
45
62 \times 45
62
×
45
E
62
×
45
62 \times 45
62
×
45
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Select the expressions that are equivalent to
6
d
+
6
d
6d + 6d
6
d
+
6
d
.
\newline
Multi-select Choices:
\newline
(A)
7
d
+
5
d
7d + 5d
7
d
+
5
d
\newline
(B)
d
×
12
d \times 12
d
×
12
\newline
(C)
12
d
12d
12
d
\newline
(D)
d
12
d^{12}
d
12
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Verify the expression:
8
(
n
!
)
(
(
m
−
3
)
!
(
n
−
m
−
3
)
!
)
=
(
m
!
)
(
(
m
−
3
)
!
(
m
−
m
−
3
)
!
)
\frac{8(n!)}{((m-3)!(n-m-3)!)}=\frac{(m!)}{((m-3)!(m-m-3)!)}
((
m
−
3
)!
(
n
−
m
−
3
)!)
8
(
n
!)
=
((
m
−
3
)!
(
m
−
m
−
3
)!)
(
m
!)
,
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Select the expressions that are equivalent to
2
(
3
g
)
2(3g)
2
(
3
g
)
.
\newline
Multi-select Choices:
\newline
(A)
2
(
g
+
2
g
)
2(g + 2g)
2
(
g
+
2
g
)
\newline
(B)
(
3
g
)
2
(3g)^2
(
3
g
)
2
\newline
(C)
6
g
6g
6
g
\newline
(D)
3
(
g
×
2
)
3(g \times 2)
3
(
g
×
2
)
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Select all the expressions that are equivalent to
1
0
4
1
0
4
\frac{10^4}{10^4}
1
0
4
1
0
4
.
\newline
Multi-select Choices:
\newline
(A)
10
10
10
\newline
(B)
1
0
8
10^8
1
0
8
\newline
(C)
1
1
1
\newline
(D)
1
0
−
6
×
1
0
6
10^{-6} \times 10^6
1
0
−
6
×
1
0
6
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Select all the expressions that are equivalent to
1
0
2
×
1
0
5
10^2 \times 10^5
1
0
2
×
1
0
5
.
\newline
Multi-select Choices:
\newline
(A)
1
0
7
10^7
1
0
7
\newline
(B)
1
1
0
7
\frac{1}{10^7}
1
0
7
1
\newline
(C)
1
1
0
−
7
\frac{1}{10^{-7}}
1
0
−
7
1
\newline
(D)
1
1
0
10
\frac{1}{10^{10}}
1
0
10
1
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The U.S. population in
1970
1970
1970
was
203
,
211
,
926
203,211,926
203
,
211
,
926
. What was the approximate population written in scientific notation?
\newline
Choices:
\newline
(A)
2
×
1
0
7
2 \times 10^7
2
×
1
0
7
\newline
(B)
2
×
1
0
8
2 \times 10^8
2
×
1
0
8
\newline
(C)
2
×
1
0
9
2 \times 10^9
2
×
1
0
9
\newline
(D)
2
×
1
0
10
2 \times 10^{10}
2
×
1
0
10
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A new shopping mall records
150
150
150
total shoppers on their first day of business. Each day after that, the number of shoppers is
15
%
15\%
15%
more than the number of shoppers the day before.
\newline
Which expression gives the total number of shoppers in the first
n
n
n
days of business?
\newline
Choose
1
1
1
answer:
\newline
(A)
1.15
(
1
−
15
0
n
1
−
150
)
1.15\left(\frac{1-150^n}{1-150}\right)
1.15
(
1
−
150
1
−
15
0
n
)
\newline
(B)
0.85
(
1
−
15
0
n
1
−
150
)
0.85\left(\frac{1-150^n}{1-150}\right)
0.85
(
1
−
150
1
−
15
0
n
)
\newline
(C)
150
(
1
−
1.1
5
n
1
−
1.15
)
150\left(\frac{1-1.15^n}{1-1.15}\right)
150
(
1
−
1.15
1
−
1.1
5
n
)
\newline
(D)
150
(
1
−
0.8
5
n
1
−
0.85
)
150\left(\frac{1-0.85^n}{1-0.85}\right)
150
(
1
−
0.85
1
−
0.8
5
n
)
Get tutor help
Divide by using long division.
\newline
(
10
x
2
+
9
x
−
1
)
÷
(
2
x
−
1
)
\left(10 x^{2}+9 x-1\right) \div(2 x-1)
(
10
x
2
+
9
x
−
1
)
÷
(
2
x
−
1
)
Get tutor help
Select all the expressions that are equivalent to
8
3
×
8
4
8^3 \times 8^4
8
3
×
8
4
.
\newline
Multi-select Choices:
\newline
(A)
1
8
−
7
\frac{1}{8^{-7}}
8
−
7
1
\newline
(B)
1
8
7
\frac{1}{8^7}
8
7
1
\newline
(C)
8
7
8^7
8
7
\newline
(D)
1
8
12
\frac{1}{8^{12}}
8
12
1
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Ellie, a teacher's assistant, is grading quizzes. The function
f
(
x
)
f(x)
f
(
x
)
gives the number of quizzes she can grade in
x
x
x
minutes.
\newline
What does
f
(
30
)
=
15
f(30) = 15
f
(
30
)
=
15
tell you?
\newline
Choices:
\newline
(A) Ellie can grade
30
30
30
quizzes in
15
15
15
minutes.
\newline
(B) Ellie can grade
15
15
15
quizzes in
30
30
30
minutes.
Get tutor help
Rose, a teacher's assistant, is grading quizzes. The function
f
(
x
)
f(x)
f
(
x
)
gives the number of quizzes she can grade in
x
x
x
minutes.
\newline
What does
f
(
30
)
=
15
f(30) = 15
f
(
30
)
=
15
tell you?
\newline
Choices:
\newline
(A) Rose can grade
15
15
15
quizzes in
30
30
30
minutes.
\newline
(B) Rose can grade
30
30
30
quizzes in
15
15
15
minutes.
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Kathleen, a teacher's assistant, is grading quizzes. The function
f
(
x
)
f(x)
f
(
x
)
gives the number of quizzes she can grade in
x
x
x
minutes.
\newline
What does
f
(
30
)
=
15
f(30) = 15
f
(
30
)
=
15
tell you?
\newline
Choices:
\newline
(A) Kathleen can grade
30
30
30
quizzes in
15
15
15
minutes.
\newline
(B) Kathleen can grade
15
15
15
quizzes in
30
30
30
minutes.
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Given the definitions of
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
below, find the value of
(
g
∘
f
)
(
4
)
(g \circ f)(4)
(
g
∘
f
)
(
4
)
.
\newline
f
(
x
)
=
2
x
2
−
2
x
−
15
g
(
x
)
=
3
x
−
4
\begin{array}{l} f(x)=2 x^{2}-2 x-15 \\ g(x)=3 x-4 \end{array}
f
(
x
)
=
2
x
2
−
2
x
−
15
g
(
x
)
=
3
x
−
4
\newline
Answer:
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The formula for the time it takes to travel a distance of
d
d
d
at an average speed
v
v
v
is
d
v
\frac{d}{v}
v
d
.
\newline
Tessa ran a
100
100
100
-meter dash at a speed of about
7
7
7
.
5
5
5
meters per second.
\newline
What calculation will give us the estimated duration of Tessa's dash in seconds?
\newline
Choose
1
1
1
answer:
\newline
(A)
7.5
100
\frac{7.5}{100}
100
7.5
\newline
(B)
100
7.5
\frac{100}{7.5}
7.5
100
\newline
(C)
7.5
100
×
60
\frac{7.5}{100\times 60}
100
×
60
7.5
\newline
(D)
100
×
60
7.5
\frac{100\times 60}{7.5}
7.5
100
×
60
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Simplify.
\newline
(
20
m
9
n
−
5
p
7
15
m
−
4
n
−
2
p
−
7
)
−
1
\left(\frac{20 m^{9} n^{-5} p^{7}}{15 m^{-4} n^{-2} p^{-7}}\right)^{-1}
(
15
m
−
4
n
−
2
p
−
7
20
m
9
n
−
5
p
7
)
−
1
\newline
Write your answer using only positive exponents.
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James had a peach that was
98
mm
98\text{mm}
98
mm
in diameter. One day he watered it with a magical solution, and it grew to
188
,
869
mm
188,869\text{mm}
188
,
869
mm
in diameter. Approximately how many times as large did the diameter of the peach become after James watered it?
\newline
Choose
1
1
1
answer:
\newline
(A)
2
×
1
0
3
2\times10^{3}
2
×
1
0
3
\newline
(B)
9
×
1
0
3
9\times10^{3}
9
×
1
0
3
\newline
(C)
2
×
1
0
4
2\times10^{4}
2
×
1
0
4
\newline
(D)
9
×
1
0
4
9\times10^{4}
9
×
1
0
4
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Solved:
\newline
Integrate
3
1
+
x
2
\frac{3}{1+x^2}
1
+
x
2
3
for a limit
[
0
,
1
]
[0,1]
[
0
,
1
]
Get tutor help
Which of the following options has the same value as
20
%
20\%
20%
of
45
45
45
?
\newline
Choose
3
3
3
answers:
\newline
(A)
20
45
×
100
\frac{20}{45}\times100
45
20
×
100
\newline
(B)
1
5
×
45
\frac{1}{5}\times45
5
1
×
45
\newline
(C)
20
×
45
20\times45
20
×
45
\newline
(D)
20
100
×
45
\frac{20}{100}\times45
100
20
×
45
\newline
(E)
0.2
×
45
0.2\times45
0.2
×
45
Get tutor help
Which expressions are equivalent to
\newline
6
×
6
×
6
×
6
×
6
6 \times 6 \times 6 \times 6 \times 6
6
×
6
×
6
×
6
×
6
?
\newline
Choose
2
2
2
answers:
\newline
A)
(
6
2
)
3
(6^{2})^{3}
(
6
2
)
3
\newline
B)
2
5
×
3
5
2^{5} \times 3^{5}
2
5
×
3
5
\newline
C)
6
6
6
1
\frac{6^{6}}{6^{1}}
6
1
6
6
\newline
D)
3
2
×
2
3
3^{2} \times 2^{3}
3
2
×
2
3
Get tutor help
The formula for the distance traveled over time
t
t
t
and at an average speed
v
v
v
is
v
⋅
t
v \cdot t
v
⋅
t
. Amit ran for
40
40
40
minutes at a speed of about
5
5
5
kilometers per hour. What calculation will give us the estimated distance Amit ran in kilometers? Choose
1
1
1
answer:
\newline
(A)
5
⋅
40
⋅
60
5 \cdot 40 \cdot 60
5
⋅
40
⋅
60
\newline
(B)
5
⋅
1000
⋅
40
5 \cdot 1000 \cdot 40
5
⋅
1000
⋅
40
\newline
(C)
5
⋅
(
40
60
)
5 \cdot \left(\frac{40}{60}\right)
5
⋅
(
60
40
)
\newline
(D)
(
5
1000
)
⋅
40
\left(\frac{5}{1000}\right) \cdot 40
(
1000
5
)
⋅
40
Get tutor help
Which expressions are equivalent to
7
7
⋅
7
7
⋅
7
7
⋅
7
7
⋅
7
7
⋅
7
7
?
7^7\cdot7^7\cdot7^7\cdot7^7\cdot7^7\cdot7^7?
7
7
⋅
7
7
⋅
7
7
⋅
7
7
⋅
7
7
⋅
7
7
?
\newline
Choose
2
2
2
answers:
\newline
(A)
7
8
7
2
\frac{7^8}{7^2}
7
2
7
8
\newline
(B)
7
6
⋅
7
1
7^6\cdot7^1
7
6
⋅
7
1
\newline
(C)
(
7
2
)
3
\left(7^2\right)^3
(
7
2
)
3
\newline
(D)
7
12
7
2
\frac{7^{12}}{7^2}
7
2
7
12
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Simplify
5
p
2
p
2
+
2
p
−
15
⋅
p
2
−
25
25
p
\frac{5 p^{2}}{p^{2}+2 p-15} \cdot \frac{p^{2}-25}{25 p}
p
2
+
2
p
−
15
5
p
2
⋅
25
p
p
2
−
25
\newline
Get tutor help
Which recursive sequence would produce the sequence
7
,
33
,
163
,
…
7,33,163, \ldots
7
,
33
,
163
,
…
?
\newline
a
1
=
7
a_{1}=7
a
1
=
7
and
a
n
=
5
a
n
−
1
−
2
a_{n}=5 a_{n-1}-2
a
n
=
5
a
n
−
1
−
2
\newline
a
1
=
7
a_{1}=7
a
1
=
7
and
a
n
=
5
a
n
−
1
+
4
a_{n}=5 a_{n-1}+4
a
n
=
5
a
n
−
1
+
4
\newline
a
1
=
7
a_{1}=7
a
1
=
7
and
a
n
=
4
a
n
−
1
+
5
a_{n}=4 a_{n-1}+5
a
n
=
4
a
n
−
1
+
5
\newline
a
1
=
7
a_{1}=7
a
1
=
7
and
a
n
=
−
2
a
n
−
1
+
5
a_{n}=-2 a_{n-1}+5
a
n
=
−
2
a
n
−
1
+
5
Get tutor help
Three points on the graph of the function
f
(
x
)
f(x)
f
(
x
)
are
{
(
0
,
2
)
,
(
1
,
3
)
,
(
2
,
4
)
}
\{(0,2),(1,3),(2,4)\}
{(
0
,
2
)
,
(
1
,
3
)
,
(
2
,
4
)}
. Which equation represents
f
(
x
)
f(x)
f
(
x
)
?
\newline
f
(
x
)
=
x
+
2
f(x)=x+2
f
(
x
)
=
x
+
2
\newline
f
(
x
)
=
x
−
2
f(x)=x-2
f
(
x
)
=
x
−
2
\newline
f
(
x
)
=
2
⋅
(
3
2
)
x
f(x)=2 \cdot\left(\frac{3}{2}\right)^{x}
f
(
x
)
=
2
⋅
(
2
3
)
x
\newline
f
(
x
)
=
x
2
+
2
f(x)=x^{2}+2
f
(
x
)
=
x
2
+
2
Get tutor help
Evaluate
6
+
4
a
+
b
3
6+\frac{4}{a}+\frac{b}{3}
6
+
a
4
+
3
b
when
a
=
4
a=4
a
=
4
and
b
=
3
b=3
b
=
3
.
Get tutor help
Which expressions are equivalent to
\newline
7
7
⋅
7
7
⋅
7
7
7^7\cdot7^7\cdot7^7
7
7
⋅
7
7
⋅
7
7
?
\newline
Choose
2
2
2
answers:
\newline
(A)
7
8
7
2
\frac{7^8}{7^2}
7
2
7
8
\newline
(B)
7
6
⋅
7
1
7^6\cdot7^1
7
6
⋅
7
1
\newline
(C)
(
7
2
)
3
(7^2)^3
(
7
2
)
3
\newline
(D)
7
12
7
2
\frac{7^{12}}{7^2}
7
2
7
12
Get tutor help
The number of nano-related patents that are granted in the US increases by a factor of
1
1
1
.
2
2
2
every year. In
1991
1991
1991
, there were
60
60
60
nano-related patents.
\newline
Which expression gives the number of patents in
1998
?
1998 ?
1998
?
\newline
Choose
1
1
1
answer:
\newline
(A)
60
+
(
1
+
1.2
)
7
60+(1+1.2)^{7}
60
+
(
1
+
1.2
)
7
\newline
(B)
60
⋅
1.
2
7
60 \cdot 1.2^{7}
60
⋅
1.
2
7
\newline
(C)
60
+
1.
2
7
60+1.2^{7}
60
+
1.
2
7
\newline
(D)
60
⋅
(
1
+
1.2
)
7
60 \cdot(1+1.2)^{7}
60
⋅
(
1
+
1.2
)
7
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Moore's law says that the number of transistors in a dense integrated circuit increases by
41
%
41 \%
41%
every year. In
1974
1974
1974
, a dense integrated circuit was produced with
5000
5000
5000
transistors.
\newline
Which expression gives the number of transistors in a dense integrated circuit in
1979
?
1979 ?
1979
?
\newline
Choose
1
1
1
answer:
\newline
(A)
5000
⋅
0.4
1
5
5000 \cdot 0.41^{5}
5000
⋅
0.4
1
5
\newline
(B)
5000
+
(
1
+
0.41
)
5
5000+(1+0.41)^{5}
5000
+
(
1
+
0.41
)
5
\newline
(C)
5000
(
1
+
0.41
)
5
5000(1+0.41)^{5}
5000
(
1
+
0.41
)
5
\newline
(D)
5000
+
0.4
1
5
5000+0.41^{5}
5000
+
0.4
1
5
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Consider the following problem:
\newline
The population of ants in Chloe's ant farm changes at a rate of
r
(
t
)
=
−
15.8
⋅
0.
9
t
r(t)=-15.8 \cdot 0.9^{t}
r
(
t
)
=
−
15.8
⋅
0.
9
t
ants per month (where
t
t
t
is time in months). At time
t
=
0
t=0
t
=
0
, the ant farm's population is
150
150
150
ants. How many ants are in the farm at
t
=
4
t=4
t
=
4
?
\newline
Which expression can we use to solve the problem?
\newline
Choose
1
1
1
answer:
\newline
(A)
∫
0
4
r
(
t
)
d
t
\int_{0}^{4} r(t) d t
∫
0
4
r
(
t
)
d
t
\newline
(B)
∫
3
4
r
(
t
)
d
t
\int_{3}^{4} r(t) d t
∫
3
4
r
(
t
)
d
t
\newline
(C)
150
+
∫
3
4
r
(
t
)
d
t
150+\int_{3}^{4} r(t) d t
150
+
∫
3
4
r
(
t
)
d
t
\newline
(D)
150
+
∫
0
4
r
(
t
)
d
t
150+\int_{0}^{4} r(t) d t
150
+
∫
0
4
r
(
t
)
d
t
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