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Math Problems
Grade 8
Multiply and divide powers: variable bases
Simplify. Express your answer using positive exponents.
\newline
10
q
5
q
⋅
q
3
\frac{10q^5}{q \cdot q^3}
q
⋅
q
3
10
q
5
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Simplify. Express your answer using positive exponents.
\newline
(
3
r
7
)
(
8
r
5
)
(
5
r
)
(3r^7)(8r^5)(5r)
(
3
r
7
)
(
8
r
5
)
(
5
r
)
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Simplify. Express your answer using positive exponents.
\newline
5
t
(
5
t
)
(
t
5
)
\frac{5t}{(5t)(t^5)}
(
5
t
)
(
t
5
)
5
t
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Simplify. Express your answer using positive exponents.
\newline
4
g
3
2
g
⋅
g
7
\frac{4g^3}{2g \cdot g^7}
2
g
⋅
g
7
4
g
3
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Simplify. Express your answer using positive exponents.
\newline
(
6
d
)
(
2
d
7
)
(
6
d
)
(6d)(2d^7)(6d)
(
6
d
)
(
2
d
7
)
(
6
d
)
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Simplify. Express your answer using positive exponents.
\newline
(
6
g
8
)
(
7
g
3
)
(
4
g
3
)
(6g^8)(7g^3)(4g^3)
(
6
g
8
)
(
7
g
3
)
(
4
g
3
)
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Simplify. Express your answer using positive exponents.
\newline
6
p
(
6
p
2
)
(
p
)
\frac{6p}{(6p^2)(p)}
(
6
p
2
)
(
p
)
6
p
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Simplify. Express your answer using positive exponents.
\newline
9
f
9
f
⋅
f
7
\frac{9f}{9f \cdot f^7}
9
f
⋅
f
7
9
f
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Simplify. Express your answer using positive exponents.
\newline
(
10
t
8
)
(
9
t
)
(
7
t
8
)
(10t^8)(9t)(7t^8)
(
10
t
8
)
(
9
t
)
(
7
t
8
)
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Simplify. Express your answer using positive exponents.
\newline
9
t
3
t
⋅
t
8
\frac{9t}{3t \cdot t^8}
3
t
⋅
t
8
9
t
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Simplify. Express your answer using positive exponents.
\newline
2
b
(
2
b
)
(
b
9
)
\frac{2b}{(2b)(b^9)}
(
2
b
)
(
b
9
)
2
b
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Simplify. Express your answer using positive exponents.
\newline
10
f
6
⋅
8
f
2
f
9
\frac{10f^6 \cdot 8f}{2f^9}
2
f
9
10
f
6
⋅
8
f
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Simplify. Express your answer using positive exponents.
\newline
(
9
k
4
)
(
7
k
4
)
(
5
k
8
)
(9k^4)(7k^4)(5k^8)
(
9
k
4
)
(
7
k
4
)
(
5
k
8
)
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Simplify. Express your answer using positive exponents.
\newline
(
8
m
4
)
(
5
m
)
(
4
m
)
(8m^4)(5m)(4m)
(
8
m
4
)
(
5
m
)
(
4
m
)
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Simplify. Express your answer using positive exponents.
\newline
8
a
7
(
4
a
6
)
(
a
5
)
\frac{8a^7}{(4a^6)(a^5)}
(
4
a
6
)
(
a
5
)
8
a
7
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Simplify. Express your answer using positive exponents.
\newline
(
2
k
9
)
(
10
k
8
)
(
9
k
2
)
(2k^9)(10k^8)(9k^2)
(
2
k
9
)
(
10
k
8
)
(
9
k
2
)
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Simplify. Express your answer using positive exponents.
\newline
8
d
2
2
d
⋅
d
2
\frac{8d^2}{2d \cdot d^2}
2
d
⋅
d
2
8
d
2
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Simplify. Express your answer using positive exponents.
\newline
(
4
u
6
)
(
5
u
7
)
(
8
u
)
(4u^6)(5u^7)(8u)
(
4
u
6
)
(
5
u
7
)
(
8
u
)
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Simplify. Express your answer using positive exponents.
\newline
10
s
(
10
s
2
)
(
s
)
\frac{10s}{(10s^2)(s)}
(
10
s
2
)
(
s
)
10
s
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Simplify. Express your answer using positive exponents.
\newline
6
j
8
(
3
j
)
(
j
)
\frac{6j^8}{(3j)(j)}
(
3
j
)
(
j
)
6
j
8
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Simplify. Express your answer using positive exponents.
\newline
(
10
n
4
)
(
4
n
4
)
(
5
n
8
)
(10n^4)(4n^4)(5n^8)
(
10
n
4
)
(
4
n
4
)
(
5
n
8
)
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Simplify. Express your answer using positive exponents.
\newline
(
8
s
8
)
(
2
s
)
s
\frac{(8s^8)(2s)}{s}
s
(
8
s
8
)
(
2
s
)
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Simplify. Express your answer using positive exponents.
\newline
4
p
9
4
p
⋅
p
\frac{4p^9}{4p \cdot p}
4
p
⋅
p
4
p
9
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Simplify. Express your answer using positive exponents.
\newline
(
4
k
8
)
(
3
k
7
)
(
6
k
5
)
(4k^8)(3k^7)(6k^5)
(
4
k
8
)
(
3
k
7
)
(
6
k
5
)
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Simplify. Express your answer using positive exponents.
\newline
9
y
2
(
3
y
)
(
y
5
)
\frac{9y^2}{(3y)(y^5)}
(
3
y
)
(
y
5
)
9
y
2
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Simplify. Express your answer using positive exponents.
\newline
(
8
a
3
)
(
3
a
6
)
(
5
a
6
)
(8a^3)(3a^6)(5a^6)
(
8
a
3
)
(
3
a
6
)
(
5
a
6
)
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Simplify. Express your answer using positive exponents.
\newline
7
m
2
7
m
9
⋅
m
\frac{7m^2}{7m^9 \cdot m}
7
m
9
⋅
m
7
m
2
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(a) Use the principle of mathematical induction to show that for all integers
n
≥
1
n \geq 1
n
≥
1
,
1
×
2
+
2
×
5
+
3
×
8
+
⋯
+
n
(
3
n
−
1
)
=
n
2
(
n
+
1
)
.
1\times2+2\times5+3\times8+\cdots+n(3n-1)=n^{2}(n+1).
1
×
2
+
2
×
5
+
3
×
8
+
⋯
+
n
(
3
n
−
1
)
=
n
2
(
n
+
1
)
.
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Find
d
y
d
x
\frac{dy}{dx}
d
x
d
y
by implicit differentiation
\newline
tan
(
4
x
+
y
)
=
4
x
\tan(4x+y)=4x
tan
(
4
x
+
y
)
=
4
x
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Can this differential equation be solved using separation of variables?
\newline
d
y
d
x
=
2
x
−
3
y
+
6
\frac{d y}{d x}=2 x-3 y+6
d
x
d
y
=
2
x
−
3
y
+
6
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B) No
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Let
g
(
x
)
=
10
x
x
3
+
5
x
g(x)=\frac{10 x}{x^{3}+5 x}
g
(
x
)
=
x
3
+
5
x
10
x
when
x
≠
0
x \neq 0
x
=
0
.
\newline
g
g
g
is continuous for all real numbers.
\newline
Find
g
(
0
)
g(0)
g
(
0
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
5
5
5
\newline
(B)
10
10
10
\newline
(C)
0
0
0
\newline
(D)
2
2
2
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