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Math Problems
Algebra 1
Multiplication with rational exponents
Simplifying the Expression:
2
1
2
×
2
3
4
×
3
×
3
3
2
2
1
2
×
3
−
1
2
\frac{2^{\frac{1}{2}}\times2^{\frac{3}{4}}\times3\times3^{\frac{3}{2}}}{2^{\frac{1}{2}}\times3^{-\frac{1}{2}}}
2
2
1
×
3
−
2
1
2
2
1
×
2
4
3
×
3
×
3
2
3
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Simplifying the Expression:
2
−
1
+
5
×
4
−
(
5
−
1
)
2-1+5\times 4-(5-1)
2
−
1
+
5
×
4
−
(
5
−
1
)
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(
t
+
8
3
)
(
t
+
b
)
=
0
(t+\frac{8}{3})(t+b)=0
(
t
+
3
8
)
(
t
+
b
)
=
0
\newline
In the given equation,
b
b
b
is a constant.
\newline
If
−
8
3
-\frac{8}{3}
−
3
8
and
13
3
\frac{13}{3}
3
13
are solutions to the equation, then what is the value of
b
b
b
?
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Solve for
x
x
x
.
\newline
9
=
−
6
x
9=-\frac{6}{x}
9
=
−
x
6
\newline
Simplify your answer as much as possible.
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Evaluate:
20
−
1
3
∗
(
−
5
)
−
20 - \frac{1}{3} * ( - 5) -
20
−
3
1
∗
(
−
5
)
−
−
36
−
9
\frac{- 36}{- 9}
−
9
−
36
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Simplify each complex fraction.
(
3
4
x
+
1
x
)
/
(
2
x
−
1
6
x
)
\left(\frac{3}{4x}+\frac{1}{x}\right)/\left(\frac{2}{x}-\frac{1}{6x}\right)
(
4
x
3
+
x
1
)
/
(
x
2
−
6
x
1
)
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Find
d
d
x
(
2
x
x
2
+
3
)
\frac{d}{dx}\left(\frac{2x}{\sqrt{x^{2}+3}}\right)
d
x
d
(
x
2
+
3
2
x
)
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Paolo helped in the community garden for
2
3
4
2\frac{3}{4}
2
4
3
hours this week. That was
1
5
6
1\frac{5}{6}
1
6
5
equal-length shifts, because Paolo stopped early one day when it started to rain.
\newline
How long is a single shift?
\newline
□
\square
□
hours
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3
3
3
markers cost
$
5.79
\$5.79
$5.79
.
\newline
Which equation would help determine the cost of
13
13
13
markers?
\newline
Choose
1
1
1
answer:
\newline
(A)
13
$
5.79
=
x
3
\frac{13}{\$5.79} = \frac{x}{3}
$5.79
13
=
3
x
\newline
(B)
x
13
=
3
$
5.79
\frac{x}{13} = \frac{3}{\$5.79}
13
x
=
$5.79
3
\newline
(C)
3
$
5.79
=
13
x
\frac{3}{\$5.79} = \frac{13}{x}
$5.79
3
=
x
13
\newline
(D)
13
x
=
$
5.79
3
\frac{13}{x} = \frac{\$5.79}{3}
x
13
=
3
$5.79
\newline
(E) None of the above
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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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Let
h
(
x
)
=
x
ln
(
x
)
h(x)=\sqrt{x} \ln (x)
h
(
x
)
=
x
ln
(
x
)
.
\newline
Find
h
′
(
x
)
h^{\prime}(x)
h
′
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
ln
(
x
)
2
x
+
1
x
\frac{\ln (x)}{2 \sqrt{x}}+\frac{1}{\sqrt{x}}
2
x
l
n
(
x
)
+
x
1
\newline
(B)
1
2
x
+
1
x
\frac{1}{2 \sqrt{x}}+\frac{1}{x}
2
x
1
+
x
1
\newline
(C)
1
2
x
⋅
1
x
\frac{1}{2 \sqrt{x}} \cdot \frac{1}{x}
2
x
1
⋅
x
1
\newline
(D)
x
ln
(
x
)
2
+
1
x
\frac{\sqrt{x} \ln (x)}{2}+\frac{1}{\sqrt{x}}
2
x
l
n
(
x
)
+
x
1
Get tutor help
Let
y
=
x
4
ln
(
x
)
y=x^{4} \ln (x)
y
=
x
4
ln
(
x
)
.
\newline
Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Choose
1
1
1
answer:
\newline
(A)
x
3
(
4
ln
(
x
)
+
1
)
x^{3}(4 \ln (x)+1)
x
3
(
4
ln
(
x
)
+
1
)
\newline
(B)
4
x
2
4 x^{2}
4
x
2
\newline
(C)
4
x
3
+
1
x
4 x^{3}+\frac{1}{x}
4
x
3
+
x
1
\newline
(D)
4
x
3
(
x
4
+
ln
(
x
)
)
4 x^{3}\left(x^{4}+\ln (x)\right)
4
x
3
(
x
4
+
ln
(
x
)
)
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Let
y
=
x
cos
(
x
)
y=\sqrt{x} \cos (x)
y
=
x
cos
(
x
)
.
\newline
Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
x
+
sin
(
x
)
-\frac{1}{\sqrt{x}}+\sin (x)
−
x
1
+
sin
(
x
)
\newline
(B)
cos
(
x
)
2
x
−
x
sin
(
x
)
\frac{\cos (x)}{2 \sqrt{x}}-\sqrt{x} \sin (x)
2
x
c
o
s
(
x
)
−
x
sin
(
x
)
\newline
(C)
−
sin
(
x
)
x
-\frac{\sin (x)}{\sqrt{x}}
−
x
s
i
n
(
x
)
\newline
(D)
−
2
x
cos
(
x
)
−
x
sin
(
x
)
-2 \sqrt{x} \cos (x)-\sqrt{x} \sin (x)
−
2
x
cos
(
x
)
−
x
sin
(
x
)
Get tutor help
d
d
x
(
5
x
ln
(
x
)
)
=
\frac{d}{d x}\left(\frac{5}{x} \ln (x)\right)=
d
x
d
(
x
5
ln
(
x
)
)
=
Get tutor help
Let
h
(
x
)
=
x
ln
(
x
)
h(x)=\sqrt{x} \ln (x)
h
(
x
)
=
x
ln
(
x
)
.
\newline
Find
h
′
(
x
)
h^{\prime}(x)
h
′
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
x
+
1
x
\frac{1}{2 \sqrt{x}}+\frac{1}{x}
2
x
1
+
x
1
\newline
(B)
ln
(
x
)
2
x
+
1
x
\frac{\ln (x)}{2 \sqrt{x}}+\frac{1}{\sqrt{x}}
2
x
l
n
(
x
)
+
x
1
\newline
(C)
1
2
x
⋅
1
x
\frac{1}{2 \sqrt{x}} \cdot \frac{1}{x}
2
x
1
⋅
x
1
\newline
(D)
x
ln
(
x
)
2
+
1
x
\frac{\sqrt{x} \ln (x)}{2}+\frac{1}{\sqrt{x}}
2
x
l
n
(
x
)
+
x
1
Get tutor help
Let
y
=
x
4
ln
(
x
)
y=x^{4} \ln (x)
y
=
x
4
ln
(
x
)
.
\newline
Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Choose
1
1
1
answer:
\newline
(A)
4
x
2
4 x^{2}
4
x
2
\newline
(B)
x
3
(
4
ln
(
x
)
+
1
)
x^{3}(4 \ln (x)+1)
x
3
(
4
ln
(
x
)
+
1
)
\newline
(C)
4
x
3
+
1
x
4 x^{3}+\frac{1}{x}
4
x
3
+
x
1
\newline
(D)
4
x
3
(
x
4
+
ln
(
x
)
)
4 x^{3}\left(x^{4}+\ln (x)\right)
4
x
3
(
x
4
+
ln
(
x
)
)
Get tutor help
Simplify the expression
\newline
(
1
−
log
(
1
a
)
(
1
(
a
−
b
)
2
)
+
log
a
2
(
a
−
b
)
)
/
(
(
1
−
log
a
(
a
−
b
)
+
log
a
2
(
a
−
b
)
)
1
2
)
.
(1-\log_{\left(\frac{1}{a}\right)}\left(\frac{1}{(a-b)^{2}}\right)+\log_{a}^{2}(a-b))/\left((1-\log_{\sqrt{a}}(a-b)+\log_{a}^{2}(a-b))^{\frac{1}{2}}\right).
(
1
−
lo
g
(
a
1
)
(
(
a
−
b
)
2
1
)
+
lo
g
a
2
(
a
−
b
))
/
(
(
1
−
lo
g
a
(
a
−
b
)
+
lo
g
a
2
(
a
−
b
)
)
2
1
)
.
Get tutor help
Solve for
b
b
b
. Express your answer in simplest radical form if necessary.
\newline
b
=
−
73
3
⋅
−
73
3
⋅
−
73
3
b=\sqrt[3]{-73} \cdot \sqrt[3]{-73} \cdot \sqrt[3]{-73}
b
=
3
−
73
⋅
3
−
73
⋅
3
−
73
\newline
Answer:
b
=
b=
b
=
Get tutor help
Solve for
b
b
b
. Express your answer in simplest radical form if necessary.
\newline
b
=
−
6
3
⋅
−
6
3
⋅
−
6
3
b=\sqrt[3]{-6} \cdot \sqrt[3]{-6} \cdot \sqrt[3]{-6}
b
=
3
−
6
⋅
3
−
6
⋅
3
−
6
\newline
Answer:
b
=
b=
b
=
Get tutor help
Solve for
a
a
a
. Express your answer in simplest radical form if necessary.
\newline
a
=
−
87
3
⋅
−
87
3
⋅
−
87
3
a=\sqrt[3]{-87} \cdot \sqrt[3]{-87} \cdot \sqrt[3]{-87}
a
=
3
−
87
⋅
3
−
87
⋅
3
−
87
\newline
Answer:
a
=
a=
a
=
Get tutor help
Solve the following equation for
x
x
x
. Express your answer in the simplest form.
\newline
3
+
3
2
(
−
10
x
+
6
)
=
−
(
10
x
+
2
)
+
1
3+\frac{3}{2}(-10 x+6)=-(10 x+2)+1
3
+
2
3
(
−
10
x
+
6
)
=
−
(
10
x
+
2
)
+
1
\newline
Get tutor help
Change the mixed number into improper fraction.
\newline
5
4
5
=
□
□
5 \frac{4}{5}=\frac{\square}{\square}
5
5
4
=
□
□
Get tutor help
Given the function
f
(
x
)
=
−
5
6
x
2
+
1
5
f(x)=-\frac{5}{6} x^{2}+\frac{1}{5}
f
(
x
)
=
−
6
5
x
2
+
5
1
, then what is
f
(
2
x
)
f(2 x)
f
(
2
x
)
as a simplified polynomial?
\newline
Answer:
Get tutor help
Given the function
f
(
x
)
=
x
−
3
2
f(x)=x-\frac{3}{2}
f
(
x
)
=
x
−
2
3
, then what is
f
(
x
+
2
)
f(x+2)
f
(
x
+
2
)
as a simplified polynomial?
\newline
Answer:
Get tutor help
Given the function
f
(
x
)
=
1
5
x
+
6
5
f(x)=\frac{1}{5} x+\frac{6}{5}
f
(
x
)
=
5
1
x
+
5
6
, then what is
f
(
x
−
3
)
f(x-3)
f
(
x
−
3
)
as a simplified polynomial?
\newline
Answer:
Get tutor help
Given the function
f
(
x
)
=
−
1
3
+
1
2
x
3
f(x)=-\frac{1}{3}+\frac{1}{2} x^{3}
f
(
x
)
=
−
3
1
+
2
1
x
3
, then what is
f
(
x
)
+
2
f(x)+2
f
(
x
)
+
2
as a simplified polynomial?
\newline
Answer:
Get tutor help
Given the function
f
(
x
)
=
5
2
x
−
1
6
f(x)=\frac{5}{2} x-\frac{1}{6}
f
(
x
)
=
2
5
x
−
6
1
, then what is
f
(
x
−
1
)
f(x-1)
f
(
x
−
1
)
as a simplified polynomial?
\newline
Answer:
Get tutor help
Given the function
f
(
x
)
=
−
5
3
x
+
5
2
x
3
f(x)=-\frac{5}{3} x+\frac{5}{2} x^{3}
f
(
x
)
=
−
3
5
x
+
2
5
x
3
, then what is
f
(
2
x
)
f(2 x)
f
(
2
x
)
as a simplified polynomial?
\newline
Answer:
Get tutor help
2
4
3
⋅
2
2
3
2\sqrt[3]{4} \cdot 2\sqrt[3]{2}
2
3
4
⋅
2
3
2
\newline
What is the value of the given expression?
\newline
Choose
1
1
1
answer:
\newline
(A)
2
6
3
2\sqrt[3]{6}
2
3
6
\newline
(B)
4
8
6
4\sqrt[6]{8}
4
6
8
\newline
(C)
8
8
8
\newline
(D)
16
16
16
Get tutor help
simplify this:
∫
cos
−
1
(
x
)
⋅
(
1
−
x
2
)
−
1
log
(
1
+
(
sin
(
2
x
1
−
x
2
)
π
)
)
d
x
\int \frac{\cos^{-1}(x) \cdot \sqrt{(1-x^2)^{-1}}}{\log \left( 1+ \left( \frac{\sin (2x\sqrt{1-x^2})}{\pi} \right) \right)} \, dx
∫
l
o
g
(
1
+
(
π
s
i
n
(
2
x
1
−
x
2
)
)
)
c
o
s
−
1
(
x
)
⋅
(
1
−
x
2
)
−
1
d
x
Get tutor help
Evaluate.
\newline
(
2
3
)
−
4
(
2
3
)
3
\left(\frac{2}{3}\right)^{-4}\left(\frac{2}{3}\right)^{3}
(
3
2
)
−
4
(
3
2
)
3
Get tutor help
Find the LCD of the rational expressions in the list.
\newline
7
5
x
+
15
,
2
9
x
−
36
\frac{7}{5 x+15}, \frac{2}{9 x-36}
5
x
+
15
7
,
9
x
−
36
2
\newline
The LCD (least common denominator) is
□
\square
□
(Type your answer in factored form.)
Get tutor help
Find
k
′
(
x
)
k^{\prime}(x)
k
′
(
x
)
if
k
(
x
)
=
−
5
x
3
⋅
e
−
3
x
4
+
4
x
2
k(x)=-\frac{5}{x^{3}} \cdot e^{-3 x^{4}+4 x^{2}}
k
(
x
)
=
−
x
3
5
⋅
e
−
3
x
4
+
4
x
2
.
Get tutor help
1
−
1
x
1
−
1
x
2
\frac{1-\frac{1}{x}}{1-\frac{1}{x^{2}}}
1
−
x
2
1
1
−
x
1
Get tutor help
(
7
4
×
5
8
)
+
(
7
4
×
1
2
)
\left(\frac{7}{4} \times \frac{5}{8}\right)+\left(\frac{7}{4} \times \frac{1}{2}\right)
(
4
7
×
8
5
)
+
(
4
7
×
2
1
)
Get tutor help
Simplify. Assume all variables are positive.
\newline
d
2
3
d
8
3
⋅
d
8
3
\frac{d^{\frac{2}{3}}}{d^{\frac{8}{3}} \cdot d^{\frac{8}{3}}}
d
3
8
⋅
d
3
8
d
3
2
\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
1
2
⋅
z
3
2
\frac{z^{\frac{1}{2}}}{z^{\frac{1}{2}} \cdot z^{\frac{3}{2}}}
z
2
1
⋅
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
r
7
4
r
7
4
⋅
r
7
4
\frac{r^{\frac{7}{4}}}{r^{\frac{7}{4}} \cdot r^{\frac{7}{4}}}
r
4
7
⋅
r
4
7
r
4
7
\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
______
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Simplify. Assume all variables are positive.
\newline
z
5
/
2
z
5
/
2
⋅
z
3
/
2
\frac{z^{5/2}}{z^{5/2} \cdot z^{3/2}}
z
5/2
⋅
z
3/2
z
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
______
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Simplify. Assume all variables are positive.
\newline
d
4
3
d
2
3
⋅
d
7
3
\frac{d^{\frac{4}{3}}}{d^{\frac{2}{3}} \cdot d^{\frac{7}{3}}}
d
3
2
⋅
d
3
7
d
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
______
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Simplify. Assume all variables are positive.
\newline
v
5
/
2
v
5
/
2
⋅
v
1
/
2
\frac{v^{5/2}}{v^{5/2} \cdot v^{1/2}}
v
5/2
⋅
v
1/2
v
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
______
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Simplify. Assume all variables are positive.
\newline
r
5
/
2
r
5
/
2
⋅
r
5
/
2
\frac{r^{5/2}}{r^{5/2} \cdot r^{5/2}}
r
5/2
⋅
r
5/2
r
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
______
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Simplify. Assume all variables are positive.
\newline
s
2
3
s
4
3
⋅
s
4
3
\frac{s^{\frac{2}{3}}}{s^{\frac{4}{3}} \cdot s^{\frac{4}{3}}}
s
3
4
⋅
s
3
4
s
3
2
\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
d
4
3
d
4
3
⋅
d
4
3
\frac{d^{\frac{4}{3}}}{d^{\frac{4}{3}} \cdot d^{\frac{4}{3}}}
d
3
4
⋅
d
3
4
d
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
______
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(d)
(
−
1
2
+
1
3
)
2
−
[
1
4
+
(
−
1
3
)
]
+
(
−
1
20
)
\left(-\frac{1}{2}+\frac{1}{3}\right)^{2}-\left[\frac{1}{4}+\left(-\frac{1}{3}\right)\right]+\left(-\frac{1}{20}\right)
(
−
2
1
+
3
1
)
2
−
[
4
1
+
(
−
3
1
)
]
+
(
−
20
1
)
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3
3
⋅
(
3
4
)
4
3^{3} \cdot\left(3^{4}\right)^{4}
3
3
⋅
(
3
4
)
4
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A sound wave travels through iron at a rate of
5120
m
s
5120 \frac{\mathrm{m}}{\mathrm{s}}
5120
s
m
.
\newline
At what rate does the sound wave travel in
k
m
h
\frac{\mathrm{km}}{\mathrm{h}}
h
km
?
\newline
□
k
m
h
\square \frac{\mathrm{km}}{\mathrm{h}}
□
h
km
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y
=
x
1
2
⋅
x
2
3
y=x^{\frac{1}{2}} \cdot x^{\frac{2}{3}}
y
=
x
2
1
⋅
x
3
2
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The expression
x
−
2
y
1
2
x
1
3
y
−
1
\frac{x^{-2} y^{\frac{1}{2}}}{x^{\frac{1}{3}} y^{-1}}
x
3
1
y
−
1
x
−
2
y
2
1
, where
x
>
1
x>1
x
>
1
and
y
>
1
y>1
y
>
1
, is
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5
r
3
2
r
2
\frac{\sqrt{5 r^{3}}}{\sqrt{2 r^{2}}}
2
r
2
5
r
3
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