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Simplify exponential expressions using exponent rules

\csc'(x) = -\csc(x)\cot(x)

Your answer should be in the form

p(x)+\frac{k}{x-1}

p

is a polynomial and

k

\frac{x^2+6x-4}{x-1}

-7

a

is a constant. Find the value of

a

\newline

It is given that the constant term in the expansion of

\left(2+\frac{1}{x}\right)^{2}(1-3 x)^{n^{2}}

67

n

is a positive integer. Find the value of

n

\newline

\left(4+\frac{4}{x}+\frac{1}{x^{2}}\right)\left[C_{0}^{n}+C_{d}^{n}(-3 x)+C_{2}^{n} 9\right.

Select the expressions that are equivalent to

4q + 2q

\newline

Multi-select Choices:

\newline

q^6

\newline

q + 6

\newline

6q

\newline

q + q + q + q + q + q

13^3-\frac{13^7}{26^2}

a

b

be rational numbers. Is

a \times b

rational or irrational?

\newline

1

\newline

\newline

\newline

(C) It can be either rational or irrational

rac{dy}{dx}=y^{

-1

-3

}, ext{ solve for } y

Which two expressions are equivalent to each other?

\newline

Multi-select Choices:

\newline

10^{10} \times 10^{-3}

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\frac{1}{10^{7}}

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\frac{1}{10^{-3}}

\newline

\frac{10^{7}}{10^{4}}

Select all the expressions that are equivalent to

3^{-8} \times 3^{-3}

\newline

Multi-select Choices:

\newline

3^{-11}

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3^{24}

\newline

\frac{3^{-8}}{3^{3}}

\newline

\frac{1}{3^{-11}}

Select all the expressions that are equivalent to

7^{-3} \times 7^{-3}

\newline

Multi-select Choices:

\newline

\frac{1}{7^{-6}}

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7^{-6}

\newline

\frac{1}{7^{6}}

\newline

\frac{1}{7^{9}}

Select all the expressions that are equivalent to

5^{-9} \times 5^{-5}

\newline

Multi-select Choices:

\newline

\frac{1}{5^{45}}

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\frac{1}{5^{14}}

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\frac{1}{5^{-14}}

\newline

5^{-14}

Select all the expressions that are equivalent to

9^{-1} \times 9^{-8}

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Multi-select Choices:

\newline

\frac{1}{9^8}

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\frac{1}{9^{-9}}

\newline

9^{-9}

\newline

9^8

Select all the expressions that are equivalent to

5^5 \times 5^5

\newline

Multi-select Choices:

\newline

5^{10}

\newline

\frac{1}{5^{10}}

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\frac{1}{5^{-10}}

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\frac{1}{5^{25}}

Select all the expressions that are equivalent to

(5^{-2})^{1}

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Multi-select Choices:

\newline

\frac{1}{5^{-1}}

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\frac{1}{5^{-2}}

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\frac{1}{5^{2}}

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5^{-1}

Select all the expressions that are equivalent to

7^{-2} \times 4^{-2}

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Multi-select Choices:

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\frac{1}{28^{-2}}

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28^{-2}

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\frac{1}{28^{2}}

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28^{4}

Select all the expressions that are equivalent to

(8^{-9})^0

\newline

Multi-select Choices:

\newline

8^{-9}

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\frac{1}{8^{-9}}

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\frac{1}{8^0}

\newline

8

Select all the expressions that are equivalent to

10^{-6} \times 10^{9}

\newline

Multi-select Choices:

\newline

\frac{1}{10^{-54}}

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10^{-54}

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10^{3}

\newline

\frac{1}{10^{3}}

Select all the expressions that are equivalent to

9^{-7} \times 9^{-1}

\newline

Multi-select Choices:

\newline

\frac{1}{9^8}

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\frac{1}{9^{-8}}

\newline

9^{-8}

\newline

9^7

Select all the expressions that are equivalent to

(5^{-5})^3

\newline

Multi-select Choices:

\newline

\frac{1}{5^{15}}

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\frac{1}{5^{-2}}

\newline

5^{-15}

\newline

5^{-2}

Select all the expressions that are equivalent to

5^{-5} \times 5^{-5}

\newline

Multi-select Choices:

\newline

\frac{1}{5^{25}}

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5^{25}

\newline

\frac{1}{5^{-10}}

\newline

\frac{1}{5^{10}}

Which is equal to

\frac{1}{4^4}

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\newline

\frac{1}{4^{-4}}

\newline

-4^4

\newline

(-4)^4

\newline

4^{-4}

LU

decomposition of a matrix

Select all of the true statements.

\newline

Multi-select Choices:

\newline

7 \times 10^{-3}

3.5

times as much as

2 \times 10^{-4}

\newline

4 \times 10^{-7}

0.01

times as much as

4 \times 10^{-5}

\newline

9 \times 10^{1}

3,000

times as much as

3 \times 10^{-2}

\newline

8 \times 10^{4}

3.5

0

times as much as

3.5

1

Polynomial function

h

h(x)=2x^{3}-9x^{2}+cx-6

c

is a constant. If

2x-3

is a factor of the polynomial, then what is the value of

c

\newline

\square

f(x+2)=g^{-1}(x+1)

f \circ g (1) = ?

f(x)=x^{2}(x+2)(x-2)(x-5)

x=-2, x=0, x=2

x=5

\newline

What is the sign of

f

on the interval

-2<x<5

\newline

1

\newline

f

is always positive on the interval.

\newline

f

is always negative on the interval.

\newline

f

is sometimes positive and sometimes negative on the interval.

f(x)=x(4 x+9)(x-2)(2 x-9)(x+5)

x=-5, x=-\frac{9}{4}, x=0, x=2

x=\frac{9}{2}

\newline

What is the sign of

f

on the interval

0<x<2

\newline

1

\newline

f

is always positive on the interval.

\newline

f

is always negative on the interval.

\newline

f

is sometimes positive and sometimes negative on the interval.

f(t)

9

over every unit interval in

t

f(0) = 0

\newline

Which could be a function rule for

f(t)

\newline

\newline

f(t) = -9t

\newline

f(t) = 9^t

\newline

f(t) = 9t

\newline

f(t)

-t

9

0

1

\newline

Use a truth table to determine whether the two statements are equivalent.

\newline

(r \wedge q) \vee p \text { and }(q \vee p) \wedge r

\newline

Construct the truth table.

\newline

\begin{tabular}{|c|c|c|c|c|}

\newline

p

q

r

(r \wedge q) \vee p

(q \vee p) \wedge r

\newline

T

T

T

T

T

\newline

\newline

Simplify. Express your answer using exponents.

\newline

\left(t^{-6} u^{2} v^{8}\right)^{-6}

Select all the expressions that are equivalent to

(2^{-6})^1

\newline

Multi-select Choices:

\newline

2^{-5}

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\frac{1}{2^6}

\newline

\frac{1}{2^{-5}}

\newline

2^{-6}

Select all the expressions that are equivalent to

(3^{-5})^{-1}

\newline

Multi-select Choices:

\newline

\frac{1}{3^5}

\newline

3^{-6}

\newline

\frac{1}{3^{-6}}

\newline

3^5

Select all the expressions that are equivalent to

5^{-4} \times 6^{-4}

\newline

Multi-select Choices:

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\frac{1}{30^4}

\newline

\frac{1}{30^{-4}}

\newline

30^{-4}

\newline

30^{-8}

Select all the expressions that are equivalent to

\frac{6^5}{2^5}

\newline

Multi-select Choices:

\newline

\frac{1}{3^5}

\newline

3

\newline

\frac{1}{3^{-5}}

\newline

3^5

Select all the expressions that are equivalent to

4 \times 4^7

\newline

Multi-select Choices:

\newline

\frac{1}{4^{-8}}

\newline

4^8

\newline

\frac{1}{4^8}

\newline

4^7

Select all the expressions that are equivalent to

5^{-9} \times 5^{9}

\newline

Multi-select Choices:

\newline

5^{0}

\newline

5^{-81}

\newline

1

\newline

\frac{1}{5^{0}}

Select all the expressions that are equivalent to

(8^{-3})^1

\newline

Multi-select Choices:

\newline

\frac{1}{8^3}

\newline

\frac{1}{8^{-3}}

\newline

8^{-2}

\newline

\frac{1}{8^{-2}}

Select all the expressions that are equivalent to

12^7 \times 12^{-5}

\newline

Multi-select Choices:

\newline

12^2

\newline

1/12^{-35}

\newline

1/12^{-2}

\newline

12^{-35}

Select all the expressions that are equivalent to

(7^1)^{-1}

\newline

Multi-select Choices:

\newline

\frac{1}{7}

\newline

7^0

\newline

\frac{1}{7^{-1}}

\newline

7^{-1}

Select all the expressions that are equivalent to

2^{-1} \times 2^{-4}

\newline

Multi-select Choices:

\newline

\frac{1}{2^{-5}}

\newline

\frac{1}{2^{5}}

\newline

2^{-5}

\newline

2^{4}

Select all the expressions that are equivalent to

3^{-8} \times 3^{-1}

\newline

Multi-select Choices:

\newline

\frac{1}{3^8}

\newline

3^{-9}

\newline

3^8

\newline

\frac{1}{3^{-9}}

Select all the expressions that are equivalent to

3^6 \times 3^{-9}

\newline

Multi-select Choices:

\newline

\frac{1}{3^3}

\newline

\frac{1}{3^{-54}}

\newline

\frac{1}{3^{-3}}

\newline

3^{-54}

Select all the expressions that are equivalent to

11^{-7} \times 11^{9}

\newline

Multi-select Choices:

\newline

\frac{1}{11^{-2}}

\newline

\frac{1}{11^{-63}}

\newline

11^{-63}

\newline

\frac{1}{11^{2}}

Select all the expressions that are equivalent to

10^3 \times 10^{-2}

\newline

Multi-select Choices:

\newline

\frac{1}{10^{-6}}

\newline

\frac{1}{10}

\newline

10^{-6}

\newline

10

Select all the expressions that are equivalent to

5^2 \times 5^{-5}

\newline

Multi-select Choices:

\newline

\frac{1}{5^{-10}}

\newline

5^{-3}

\newline

5^{-10}

\newline

\frac{1}{5^{-3}}

Select all the expressions that are equivalent to

\frac{7^{30}}{7^6}

\newline

Multi-select Choices:

\newline

7^{24}

\newline

\frac{1}{7^5}

\newline

7^{36}

\newline

\frac{1}{7^{-24}}

Select all the expressions that are equivalent to

5 \times 5^{-1}

\newline

Multi-select Choices:

\newline

\frac{1}{5^0}

\newline

5^{-1}

\newline

1

\newline

\frac{1}{5^{-1}}

Use Genenalized mean value twoorem to estalslish the following inequality for

x 50

\newline

\frac{1}{2(x+1) e^{2 x}}<\frac{\ln (x+1)}{e^{2 x}-1}<\frac{1}{2}

n

is a positive integer such that

n^2 - 7n + 17

is equal to the product of two consecutive odd integers, find the sum of these odd integers

5^2(10)^{-4}

...