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Math Problems
Algebra 2
Simplify radical expressions with variables II
1
1
1
. A sequence is defined by
f
(
0
)
=
−
21
,
f
(
n
)
=
f
(
n
−
1
)
−
5
f(0)=-21, f(n)=f(n-1)-5
f
(
0
)
=
−
21
,
f
(
n
)
=
f
(
n
−
1
)
−
5
for
n
≥
1
n \geq 1
n
≥
1
Explain why
\newline
f
(
1
)
=
−
21
−
5
f(1)=-21-5
f
(
1
)
=
−
21
−
5
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REQUIRED
\newline
4
4
4
. A sequence is defined by
f
(
0
)
=
−
11
,
f
(
n
)
=
f
(
n
−
1
)
−
6
f(0)=-11, f(n)=f(n-1)-6
f
(
0
)
=
−
11
,
f
(
n
)
=
f
(
n
−
1
)
−
6
for
n
≥
1
n \geq 1
n
≥
1
. Enter the explicit form of the equation as an
f
(
n
)
=
f(n)=
f
(
n
)
=
equation.
\newline
Type a response
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1
1
1
. A sequence is defined by
f
(
0
)
=
−
21
,
f
(
n
)
=
f
(
n
−
1
)
−
5
f(0)=-21, f(n)=f(n-1)-5
f
(
0
)
=
−
21
,
f
(
n
)
=
f
(
n
−
1
)
−
5
for
n
≥
1
n \geq 1
n
≥
1
\newline
Explain why
\newline
f
(
1
)
=
−
21
−
5
f(1)=-21-5
f
(
1
)
=
−
21
−
5
\newline
Show Your Work
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In rectangle
A
B
C
D
A B C D
A
BC
D
above,
A
D
=
12
,
A
B
=
6
A D=12, A B=6
A
D
=
12
,
A
B
=
6
, and
E
G
=
4
E G=4
EG
=
4
. What is the area of
△
E
F
G
\triangle E F G
△
EFG
?
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Question
12
12
12
\newline
0
/
10
0 / 10
0/10
pts
\newline
5
5
5
\newline
98
98
98
\newline
Details
\newline
Let
A
=
[
−
3
−
4
0
4
−
4
−
2
]
,
B
=
[
−
4
0
−
3
0
−
2
−
1
]
A=\left[\begin{array}{cc}-3 & -4 \\ 0 & 4 \\ -4 & -2\end{array}\right], B=\left[\begin{array}{cc}-4 & 0 \\ -3 & 0 \\ -2 & -1\end{array}\right]
A
=
⎣
⎡
−
3
0
−
4
−
4
4
−
2
⎦
⎤
,
B
=
⎣
⎡
−
4
−
3
−
2
0
0
−
1
⎦
⎤
\newline
A. It is Select an answer
0
0
0
to compute
(
A
−
5
B
)
T
(A-5 B)^{T}
(
A
−
5
B
)
T
because select an answer
\newline
B. If the computation is possible, calculate
(
A
−
5
B
)
T
(A-5 B)^{T}
(
A
−
5
B
)
T
. If the answer does not exist, enter DNE.
\newline
Note: To enter a matrix, click inside the answer box and choose the
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Use one exponent to condense the expression below. Then compute and/or simplify.
\newline
2
b
⋅
2
b
⋅
2
b
2 b \cdot 2 b \cdot 2 b
2
b
⋅
2
b
⋅
2
b
\newline
Answer Attempt
1
1
1
out of
2
2
2
\newline
Type the base, then use the
a
b
a^{b}
a
b
button or type the
∧
{ }^{\wedge}
∧
symbol on your keyboard for the exponent.
\newline
Condensed form:
□
\square
□
a
b
a^{b}
a
b
\newline
Answer:
□
\square
□
a
t
a^{t}
a
t
\newline
Submit Answer
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Use exponents to condense the expression below.
\newline
x
⋅
x
⋅
x
⋅
x
⋅
x
⋅
x
⋅
y
⋅
y
⋅
y
⋅
y
⋅
y
⋅
z
x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y \cdot z
x
⋅
x
⋅
x
⋅
x
⋅
x
⋅
x
⋅
y
⋅
y
⋅
y
⋅
y
⋅
y
⋅
z
\newline
Answer Attempt
1
1
1
out of
2
2
2
\newline
Type the base, then use the
a
b
a^{b}
a
b
button or use the
∧
{ }^{\wedge}
∧
symbol on your keyboard for the exponent.
\newline
Condensed form:
□
\square
□
a
b
a^{b}
a
b
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Let's Try It Together: Identifying Explicit Statements
\newline
Read this sentence: 'Emma always finishes her homework on time.' Is this an explicit statement?
\newline
Su respuesta
\newline
Escribe aquí...
\newline
9
9
9
\newline
G
\newline
Diapositiva
3
3
3
/
17
17
17
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In the expression below different letters stand for different one-digit numbers:
\newline
E
×
V
×
E
×
R
×
Y
×
T
×
H
×
I
×
N
×
G
×
I
×
S
×
G
×
R
×
E
×
A
×
T
E \times V \times E \times R \times Y \times T \times H \times I \times N \times G \times I \times S \times G \times R \times E \times A \times T
E
×
V
×
E
×
R
×
Y
×
T
×
H
×
I
×
N
×
G
×
I
×
S
×
G
×
R
×
E
×
A
×
T
\newline
a) What is the maximum value the expression can have?
\newline
b) What could be the maximum value, if we cross out one letter with all its repetitions?
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The tables below show some inputs and outputs of functions
\newline
g and
\newline
h.
\newline
\begin{array}{c|c} x & g(x) \ \hline 1 & 1 \ 4 & 2 \ 9 & 3 \ 16 & 4 \ 25 & 5 \ 36 & 6 \ \end{array}
\newline
\begin{array}{c|c} x & h(x) \ \hline 5 & 1 \ 10 & 2 \ 15 & 3 \ 20 & 4 \ 25 & 5 \ 30 & 6 \ \end{array}
\newline
Evaluate.
\newline
(
h
∘
g
)
(
25
)
=
(h \circ g)(25) =
(
h
∘
g
)
(
25
)
=
\newline
◻
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Rectangle
A
B
C
D
A B C D
A
BC
D
is graphed in the coordinate plane. The following are the vertices of the rectangle:
A
(
5
,
1
)
,
B
(
7
,
1
)
,
C
(
7
,
6
)
A(5,1), B(7,1), C(7,6)
A
(
5
,
1
)
,
B
(
7
,
1
)
,
C
(
7
,
6
)
, and
D
(
5
,
6
)
D(5,6)
D
(
5
,
6
)
.
\newline
What is the perimeter of rectangle
A
B
C
D
A B C D
A
BC
D
?
□
\square
□
\newline
units
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Want to learn more about order of operation
\newline
Practice
\newline
PROBLEM
2
2
2
\newline
7
2
+
3
(
4
2
+
3
+
2
)
=
7^{2}+3\left(4^{2}+3+2\right)=
7
2
+
3
(
4
2
+
3
+
2
)
=
□
\square
□
\newline
Chtrick
\newline
Explain
\newline
Want to practice more problems like these? Check ou and these more challenging exercises: exercise one an
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The given equation
a
i
m
e
s
x
2
+
98
x
+
c
a imes x^2 + 98x + c
aim
es
x
2
+
98
x
+
c
has at least
1
1
1
real root and a factor of
k
x
+
j
kx + j
k
x
+
j
. What is the greatest possible value of
a
c
ac
a
c
?
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The function
g
g
g
is defined by
g
(
x
)
=
1
m
x
+
21
g(x)=\frac{1}{m} x+21
g
(
x
)
=
m
1
x
+
21
, where
m
m
m
is an integer constant and
14
≤
m
≤
17
14 \leq m \leq 17
14
≤
m
≤
17
. For the graph of
y
=
g
(
x
)
−
9
y=g(x)-9
y
=
g
(
x
)
−
9
in the
x
y
x y
x
y
-plane, what is the
x
x
x
-coordinate of a possible
x
x
x
-intercept?
\newline
A)
−
168
-168
−
168
\newline
B)
−
192
-192
−
192
\newline
C)
−
204
-204
−
204
\newline
D)
−
216
-216
−
216
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Which table matches the rule
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Rewrite the expression in the form
k
⋅
y
n
k \cdot y^{n}
k
⋅
y
n
. Write the exponent as an integer, fraction, or an exact decimal (not a mixed number).
\newline
(
4
y
5
4
)
1
2
=
2
\left(4 \sqrt[4]{y^{5}}\right)^{\frac{1}{2}}=2
(
4
4
y
5
)
2
1
=
2
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Learn with an example
∼
\sim
∼
or Watch a video
\newline
valuate. Write your answer as a whole number or as a simplified fraction.
\newline
3
2
⋅
3
2
=
3^{2} \cdot 3^{2}=
3
2
⋅
3
2
=
\newline
□
\square
□
\newline
Submit
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Solve this system of equations by graphing. First graph the equations, and then type the solution.
y
=
1
2
x
−
3
y = \frac{1}{2} x - 3
y
=
2
1
x
−
3
y
=
−
3
2
x
+
1
y = - \frac{3}{2} x + 1
y
=
−
2
3
x
+
1
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Think like a mathematician
\newline
6
6
6
a Use a calculator to find
\newline
i
8
×
2
\quad \sqrt{8} \times \sqrt{2}
8
×
2
\newline
ii
3
×
12
\sqrt{3} \times \sqrt{12}
3
×
12
\newline
iii
20
×
5
\sqrt{20} \times \sqrt{5}
20
×
5
\newline
iv
2
×
18
\sqrt{2} \times \sqrt{18}
2
×
18
\newline
b What do you notice about your answers?
\newline
c Find another multiplication similar to the multiplications in part a.
\newline
d Find similar multiplications using cube roots instead of square roots.
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Without using a calculator, fill in the blanks with two consecutive integers to complete the following inequality.
\newline
□
\square
□
\newline
<
56
<
<\sqrt{56}<
<
56
<
\newline
□
\square
□
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The number
c
c
c
is irrational. Which statement about
c
⋅
40
c \cdot \sqrt{40}
c
⋅
40
is true?
\newline
c
⋅
40
c \cdot \sqrt{40}
c
⋅
40
is rational.
\newline
c
⋅
40
c \cdot \sqrt{40}
c
⋅
40
is irrational.
\newline
c
⋅
40
c \cdot \sqrt{40}
c
⋅
40
can be rational or irrational, depending on the value of
c
c
c
.
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6
6
6
Pulen carried out a survey of the number of books that people in his class owned. Here are his results.
\newline
\begin{tabular}{|c|c|}
\newline
\hline Number of books & Frequency \\
\newline
\hline
0
−
4
0-4
0
−
4
&
11
11
11
\\
\newline
\hline
5
−
9
5-9
5
−
9
&
9
9
9
\\
\newline
\hline
10
−
14
10-14
10
−
14
&
4
4
4
\\
\newline
\hline
15
−
19
15-19
15
−
19
&
2
2
2
\\
\newline
\hline
20
−
24
20-24
20
−
24
&
3
3
3
\\
\newline
\hline
25
−
29
25-29
25
−
29
&
2
2
2
\\
\newline
\hline
\newline
\end{tabular}
\newline
a What is the modal class?
\newline
b Copy the axes and complete the bar chart for the data. Give your chart a title.
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Choose the correct symbol to compare the expressions. Do not multiply.
\newline
9
×
2
5
9 \times \frac{2}{5}
9
×
5
2
?
9
9
9
\newline
Choices:
\newline
(A)
>
>
>
\newline
(B)
<
<
<
\newline
(C)
=
=
=
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Choose the correct symbol to compare the expressions. Do not multiply.
\newline
5
×
2
3
5 \times \frac{2}{3}
5
×
3
2
?
5
5
5
\newline
Choices:
\newline
(A)
>
>
>
\newline
(B)
<
<
<
\newline
(C)
=
=
=
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Choose the correct symbol to compare the expressions. Do not multiply.
\newline
9
×
1
3
9 \times \frac{1}{3}
9
×
3
1
?
9
9
9
\newline
Choices:
\newline
(A)
>
>
>
\newline
(B)
<
<
<
\newline
(C)
=
=
=
Get tutor help
Find the difference quotient
f
(
x
+
h
)
−
f
(
x
)
h
\frac{f(x+h)-f(x)}{h}
h
f
(
x
+
h
)
−
f
(
x
)
, where
h
≠
0
h \neq 0
h
=
0
, for the function below.
\newline
f
(
x
)
=
8
x
−
5
f(x)=8 x-5
f
(
x
)
=
8
x
−
5
\newline
Simplify your answer as much as possible.
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Find the difference quotient
f
(
x
+
h
)
−
f
(
x
)
h
\frac{f(x+h)-f(x)}{h}
h
f
(
x
+
h
)
−
f
(
x
)
, where
h
≠
0
h \neq 0
h
=
0
, for the function below.
\newline
f
(
x
)
=
6
x
−
2
f(x)=6 x-2
f
(
x
)
=
6
x
−
2
\newline
Simplify your answer as much as possible.
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3
3
3
. Determine whether each expression is equivalent to
x
5
3
x^{\frac{5}{3}}
x
3
5
. Select Yes or No for each expression.
\newline
a.
x
\sqrt{x}
x
No
\newline
b.
x
5
3
\sqrt[3]{x^{5}}
3
x
5
Yes
\newline
c.
x
3
5
\sqrt[5]{x^{3}}
5
x
3
No
\newline
d.
x
5
3
\sqrt{x^{\frac{5}{3}}}
x
3
5
No
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AP Calculus AB
\newline
AP Exam Review Free Response
6
6
6
\newline
This Question is
∗
∗
{ }^{* *}
∗∗
CALCULATOR INACTIVE**
\newline
Please show all work on page
2
&
3
2 \& 3
2&3
\newline
The equation of the implicitly defined Tschirnhausen's Cubic is
y
2
−
3
x
2
−
x
3
=
0
y^{2}-3 x^{2}-x^{3}=0
y
2
−
3
x
2
−
x
3
=
0
. The graph of Tschirnhausen's Cubic is shown below.
\newline
(a) Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
. Show all work!
\newline
(b) Find the equation of the tangent line to Tschirnhausen's Cubic at the point
(
1
,
−
2
)
(1,-2)
(
1
,
−
2
)
and use it to approximate the
y
y
y
value of the curve where
x
=
1.1
x=1.1
x
=
1.1
.
\newline
(c) At what point(s) does this curve have a horizontal tangent? Give both the
x
x
x
- and
y
y
y
-coordinates. Show all work.
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The period
T
T
T
(in seconds) of a pendulum is given by
T
=
2
π
(
L
32
)
T=2\pi\sqrt{\left(\frac{L}{32}\right)}
T
=
2
π
(
32
L
)
, where
L
L
L
stands for the length (in feet) of the pendulum
π
=
3.14
\pi=3.14
π
=
3.14
, and the period is
12.56
12.56
12.56
seconds, what is the length? The length of the pendulum is
□
\square
□
feet.
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Name Aliya buge
\newline
Period
4
4
4
\newline
40
40
40
\newline
Quiz * Modified *
\newline
1
1
1
. Fill out the t-chart about expressions and equations. (
6
6
6
points)
\newline
Expressions
\newline
1
1
1
.
N
0
=
N_{0}=
N
0
=
\newline
2
2
2
. Evaluwat
\newline
3
3
3
. varble oney one numben
\newline
Equations
\newline
1
1
1
. hasan=
\newline
2
2
2
. Solve
\newline
3
3
3
. Varble oney one nabep
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Use the statement to answer the question.
\newline
×
2
3
<
2
3
\times \frac{2}{3}<\frac{2}{3}
×
3
2
<
3
2
\newline
Which number makes the statement true?
\newline
A.
2
3
\frac{2}{3}
3
2
\newline
B.
1
1
1
\newline
C.
4
3
\frac{4}{3}
3
4
\newline
D.
3
2
\frac{3}{2}
2
3
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Select the expressions that are equivalent to
5
m
+
4
m
5m + 4m
5
m
+
4
m
.
\newline
Multi-select Choices:
\newline
(A)
m
+
9
m
m + 9m
m
+
9
m
\newline
(B)
8
m
8m
8
m
\newline
(C)
m
×
9
m \times 9
m
×
9
\newline
(D)
m
+
8
m
m + 8m
m
+
8
m
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If
r
=
∣
r
⃗
∣
r=|\vec{r}|
r
=
∣
r
∣
, where
r
⃗
=
x
ı
^
+
y
ȷ
^
+
z
k
^
\vec{r}=x \hat{\imath}+y \hat{\jmath}+z \hat{k}
r
=
x
^
+
y
^
+
z
k
^
, then prove that \vec{\(\newline\)abla} .
\newline
*** END OF PAPER ***
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Import favorites
\newline
MVNU Students Ho...
\newline
A
\newline
ALEKS - Jameson C...
\newline
Triangles and Vectors
\newline
Solving a triangle with the law of sines: Problem type
2
2
2
\newline
Jameson
\newline
Español
\newline
Consider a triangle
A
B
C
A B C
A
BC
like the one below. Suppose that
a
=
46
,
c
=
12
a=46, c=12
a
=
46
,
c
=
12
, and
A
=
9
7
∘
A=97^{\circ}
A
=
9
7
∘
. (The figure is not drawn to scale.) Solve the triangle.
\newline
Carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth.
\newline
If no such triangle exists, enter
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Use the Ratio Test to determine whether the series is convergent or divergent.
\newline
∑
k
=
1
∞
9
k
e
−
k
\sum_{k=1}^{\infty} 9 k e^{-k}
k
=
1
∑
∞
9
k
e
−
k
\newline
Identify
a
k
a_{k}
a
k
.
\newline
Evaluate the following limit.
\newline
lim
k
→
∞
∣
a
k
+
1
a
k
∣
\lim _{k \rightarrow \infty}\left|\frac{a_{k}+1}{a_{k}}\right|
k
→
∞
lim
∣
∣
a
k
a
k
+
1
∣
∣
\newline
□
\square
□
\newline
Since
lim
k
→
∞
∣
a
k
+
1
a
k
∣
?
∨
1
,
−
\lim _{k \rightarrow \infty}\left|\frac{a_{k}+1}{a_{k}}\right| ? \vee 1,-
lim
k
→
∞
∣
∣
a
k
a
k
+
1
∣
∣
?
∨
1
,
−
Select--
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Approximate the area between
h
(
x
)
h(x)
h
(
x
)
and the
x
x
x
-axis from
x
=
−
2
x=-2
x
=
−
2
to
x
=
4
x=4
x
=
4
using a right Riemann sum with
3
3
3
equal subdivisions.
\newline
R
(
3
)
=
R(3)=
R
(
3
)
=
\newline
□
\square
□
units
2
^{2}
2
.
\newline
Show Calculator
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The positive numbers
x
x
x
and
a
−
x
a-x
a
−
x
have a sum of
a
a
a
. What is
x
x
x
in terms of
a
a
a
if the product
x
⋅
(
a
−
x
)
x \cdot(a-x)
x
⋅
(
a
−
x
)
is a maximum?
\newline
Choose
1
1
1
answer:
\newline
(A)
a
2
\frac{a}{2}
2
a
\newline
(B)
a
2
\sqrt{\frac{a}{2}}
2
a
\newline
(C)
a
\sqrt{a}
a
\newline
(D)
a
a
a
\newline
(E) There is no
x
x
x
that would produce a maximum product
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Solve for
c
c
c
. Express your answer in simplest radical form if necessary.
\newline
c
=
−
98
⋅
−
98
c=-\sqrt{98} \cdot-\sqrt{98}
c
=
−
98
⋅
−
98
\newline
Answer:
c
=
c=
c
=
Get tutor help
Solve for
b
b
b
. Express your answer in simplest radical form if necessary.
\newline
b
=
86
⋅
86
b=\sqrt{86} \cdot \sqrt{86}
b
=
86
⋅
86
\newline
Answer:
b
=
b=
b
=
Get tutor help
Solve the following equation for
x
x
x
. Express your answer in the simplest form.
\newline
−
(
5
x
−
7
)
−
3
=
5
x
+
2
(
−
5
x
+
2
)
-(5 x-7)-3=5 x+2(-5 x+2)
−
(
5
x
−
7
)
−
3
=
5
x
+
2
(
−
5
x
+
2
)
\newline
Get tutor help
Simplify. Assume all variables are positive.
\newline
t
3
2
t
3
2
⋅
t
1
2
\frac{t^{\frac{3}{2}}}{t^{\frac{3}{2}} \cdot t^{\frac{1}{2}}}
t
2
3
⋅
t
2
1
t
2
3
\newline
Write your answer in the form
A
A
A
or
A
B
\frac{A}{B}
B
A
, where
A
A
A
and
B
B
B
are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
\newline
______
Get tutor help
Simplify. Assume all variables are positive.
\newline
z
1
2
z
3
2
⋅
z
3
2
\frac{z^{\frac{1}{2}}}{z^{\frac{3}{2}} \cdot z^{\frac{3}{2}}}
z
2
3
⋅
z
2
3
z
2
1
\newline
Write your answer in the form
A
A
A
or
A
B
\frac{A}{B}
B
A
, where
A
A
A
and
B
B
B
are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
\newline
______
Get tutor help
Simplify. Assume all variables are positive.
\newline
s
1
2
s
3
2
⋅
s
3
2
\frac{s^{\frac{1}{2}}}{s^{\frac{3}{2}} \cdot s^{\frac{3}{2}}}
s
2
3
⋅
s
2
3
s
2
1
\newline
Write your answer in the form
A
A
A
or
A
B
\frac{A}{B}
B
A
, where
A
A
A
and
B
B
B
are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
\newline
______
Get tutor help
Simplify. Assume all variables are positive.
\newline
s
4
3
s
2
3
⋅
s
1
3
\frac{s^{\frac{4}{3}}}{s^{\frac{2}{3}} \cdot s^{\frac{1}{3}}}
s
3
2
⋅
s
3
1
s
3
4
\newline
Write your answer in the form
A
A
A
or
A
B
\frac{A}{B}
B
A
, where
A
A
A
and
B
B
B
are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
\newline
______
Get tutor help
Simplify. Assume all variables are positive.
\newline
c
5
/
2
c
5
/
2
⋅
c
3
/
2
\frac{c^{5/2}}{c^{5/2} \cdot c^{3/2}}
c
5/2
⋅
c
3/2
c
5/2
\newline
Write your answer in the form
A
A
A
or
A
/
B
A/B
A
/
B
, where
A
A
A
and
B
B
B
are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
\newline
______
Get tutor help
Simplify. Assume all variables are positive.
\newline
w
1
3
w
4
3
⋅
w
7
3
\frac{w^{\frac{1}{3}}}{w^{\frac{4}{3}} \cdot w^{\frac{7}{3}}}
w
3
4
⋅
w
3
7
w
3
1
\newline
Write your answer in the form
A
A
A
or
A
B
\frac{A}{B}
B
A
, where
A
A
A
and
B
B
B
are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
\newline
______
Get tutor help
Simplify. Assume all variables are positive.
\newline
b
4
3
b
4
3
⋅
b
1
3
\frac{b^{\frac{4}{3}}}{b^{\frac{4}{3}} \cdot b^{\frac{1}{3}}}
b
3
4
⋅
b
3
1
b
3
4
\newline
Write your answer in the form
A
A
A
or
A
B
\frac{A}{B}
B
A
, where
A
A
A
and
B
B
B
are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
\newline
______
Get tutor help
Simplify. Assume all variables are positive.
\newline
b
3
2
b
3
2
⋅
b
5
2
\frac{b^{\frac{3}{2}}}{b^{\frac{3}{2}} \cdot b^{\frac{5}{2}}}
b
2
3
⋅
b
2
5
b
2
3
\newline
Write your answer in the form
A
A
A
or
A
B
\frac{A}{B}
B
A
, where
A
A
A
and
B
B
B
are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
\newline
______
Get tutor help
Evaluate the left hand side to find the value of
a
a
a
in the equation in simplest form.
\newline
x
4
3
x
2
3
=
x
a
\frac{x^{\frac{4}{3}}}{x^{\frac{2}{3}}}=x^{a}
x
3
2
x
3
4
=
x
a
\newline
□
\square
□
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