Simplify the following using law of indices. a) 12x⁷ ÷ 4x³
b) (2x)² × (2x)³ × (2x)⁵
​

Following are three steps 1. Isolate the variable: First, identify the variable you are trying to solve for and use the other terms in the equation to isolate that variable. This means using inverse operations to move any terms containing the variable to one side of the equation and any terms without the variable to the other side of the equation.

2. Simplify the equation: Once the variable is isolated, use the properties of equality and inverse operations to simplify the equation, if needed.

3. Solve the equation: Finally, use the inverse operations to solve the equation and find the value of the variable.

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The general solution of the given differential equation is y(x) = C₁e^-x + 4xe^-x. The largest interval over which the general solution is defined is (-∞, ∞). There are no transient terms in the general solution.

The differential equation given can be rearranged into the form dy/dx + (x + 2)/(x + 1)y = 8xe^-x, which is a linear first order differential equation. Using the integrating factor method, an integrating factor of e^(x + 1) can be used to obtain the general solution y(x) = C₁e^-x + 4xe^-x.

The integration constant C₁ can be determined by supplying the appropriate initial conditions. As the equation is a linear first order differential equation, the general solution is valid over the entire domain of x, which is (-∞, ∞). As the equation does not involve any terms that provide a particular value for y at any point in the domain, there are no transient terms in the general solution.

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The equation in option e) is not correct because it would not give correct height of the trees in this problem

Since the Magnolia trees are 3 feet tall and grow at an average of 3/4 foot per year.

If x is the number of years and y is the height. we can write:

y = 3/4(x) + 3

The slope-point equation of a line is given as:

y-y1 = m(x−x1)

y = 3/4(x) + 3 in slope-point  form is:

y = 3/4(x) + 3

y-3 =  3/4(x-0)

This means option a) is correct

Yes, this equation is okay to use for the height of the trees in this problem

b) y - 6 = 3/4(x-4)

y - 6 = 3/4(x-4)

y - 6 = 3/4(x)- 3

y - 6 + 3 = 3/4(x)

y - 3 = 3/4(x)

This is correct

c) y - 4.5 = 3/4(x-2)

y - 4.5 = 3/4(x-2)

y - 4.5 = 3/4(x)- 1.5

y - 4.5 + 1.5 = 3/4(x)

y - 3 = 3/4(x)

This is correct

d) y - 12 = 3/4(x-12)

y - 12 = 3/4(x-12)

y - 12 = 3/4(x) - 9

y - 12 + 9 = 3/4(x)

y - 3 = 3/4(x)

This is correct

(e) y - 15= 3/4(x-15)

y - 15= 3/4(x-15)

y - 15= 3/4(x)-11.25

y - 15 + 11.25= 3/4(x)

y - 3.75 = 3/4(x)

This is not correct because it does not reflect the given information

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The domain of the function f(x) = log x is (0, ∞). The value of x belongs to (0, ∞).

What is a logarithm?

The logarithm is exponentiation's opposite function in mathematics. This indicates that the exponent to which b must be raised in order to obtain a number x is the logarithm of x to the base b.

The given function is f(x) = log x.

When the value of x is less than or equal to 0, then the value of log x is undefined.

The possible value of x where f(x) exists is the domain of a function.

In the given function the value of x must be greater than zero.

The domain of the function is (0,∞).

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The coefficient of the variable x  in the expression 4a+3x +a is 3

Given that,

The expression is 4a+3x +a which can be simplified as 4a+3x +a

To find : The coefficient of the variable x in the expression

The coefficient of x in 4a+3x +a is 3

A number multiplied by a variable is referred to as the coefficient of a number. For instance, the x2 coefficient in the formula 6x2+99 is 6

Coefficient is the number that is beside the variable in the equation.

As we can see that the term that is in multiplication with x is 3

Hence, the coefficient of x in the equation  is 3

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Rounded to the nearest tenth, the measure of side ST is 8.0.

In order to find the measure of side ST, you will need to use the concept of similarity. If two triangles are similar, then their corresponding sides are in proportion. If we let x be the measure of side ST, then we can set up a proportion using the given information:

[tex]OP/PQ = ST/RQ[/tex]

We also know that

[tex]OP/PQ = 2/3[/tex]

and

[tex]RQ = 12[/tex]

So we can substitute these values into the proportion and solve for x:

[tex](2/3) = ST/12[/tex]

[tex]ST = (2/3) * 12[/tex]

[tex]ST = 8[/tex]

Rounded to the nearest tenth, the measure of side ST is 8.0.

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Answer:

The fourth one could not represent the three sides of a triangle

Step-by-step explanation:

because the two sides of the triangle must be larger together than the third side and 7+16=23<24, so the fourth one could not represent the three sides of a triangle.

Answer:

C

Step-by-step explanation: The third one down. There are two ways you know this.

1. If you had two boards that where 6 cm and 20 cm and tried to make a triangle with the third side = 27, the two little boards would lie flat on the 27 cm board.

2. The other three answers all make the smaller two sides larger than the third larger side.

By working together, they can make the work done in two hours and forty-six minutes.

Given data :

Think of these work rates as "greenhouses per hour", not "hours per greenhouse". Then Steve's work rate is one fourth of a greenhouse per hour, and his son's is one ninth of a greenhouse per hour. Now, multiply each rate by the time t to get the portion of the greenhouse each paints, and add the products up to get one greenhouse. The work equation becomes:

1 / 4 t + 1 / 9 t = 1

9 / 36 . t + 4 / 36 t = 1

13 / 36 t = 1

36 / 13. 13 / 36 t  = 36 / 13 .1

t = 36 / 13 hrs

or 36 / 13. 60 = 166 min

This is two hours and forty-six minutes.

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Answer:

decreasing is the answer

[tex](\stackrel{x_1}{4}~,~\stackrel{y_1}{-6})\qquad (\stackrel{x_2}{-2}~,~\stackrel{y_2}{3}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{3}-\stackrel{y1}{(-6)}}}{\underset{\textit{\large run}} {\underset{x_2}{-2}-\underset{x_1}{4}}} \implies \cfrac{3 +6}{-6} \implies \cfrac{ 9 }{ -6 } \implies - \cfrac{3 }{ 2 }[/tex]

Answer:

9/-6 or -1.5

Step-by-step explanation:

Follow the steps in the image

-Hope this Helped

The water level is rising at a rate of 0.5tf² /min*ft². This is in cubic feet per minute per square feet, if you want to express this in terms of height, you will have to convert the units

To find out how fast the water level is rising, we need to use the concept of volume flow rate. Volume flow rate is the volume of fluid that passes through a given surface per unit of time. In this case, the volume flow rate is given as 3tf^3/min.

Since the base of the tank is a rectangle with dimensions 2x3 ft, we can find the area of the base by multiplying the length and width:

2 ft * 3 ft = 6 ft^2

To find the rate of change of the water level, we need to divide the volume flow rate by the area of the base.

3tf^3/min / 6 ft^2 = 0.5tf^3/min*ft^2

So the water level is rising at a rate of 0.5tf^3/min*ft^2

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The equation of the line in fully simplified intersect is y = mx + b

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Therefore, amu (atomic mass unit) and Avogadro's number are not the same things.

A unit of measurement for atomic and molecular weight is the atomic mass unit (amu).

Molecular weight, commonly known as molecular mass, is the mass of a substance's molecules, calculated using the atomic weight of carbon-12, which is 12.

No, amu and Avogadro's numbers are not the same things because An atomic mass unit (amu) is a unit of measurement for atomic and molecular weight. It is defined as one twelfth of the mass of a neutral atom of carbon-12. It is commonly used in chemistry to express the masses of atoms and molecules While Avogadro's number (N) is a fundamental constant of nature that relates the number of atoms or molecules in a sample to the mass of that sample. It is defined as the number of atoms or molecules in one mole of a substance. It is approximately 6.022 x 10^23 atoms or molecules per mole.

So, amu is a unit of measurement for atomic and molecular weight, while Avogadro's number is a constant that relates the number of atoms or molecules in a sample to the mass of that sample.

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The linear expression that is not equivalent to the other three is D) (-5x-1)+(6x + 7).

A linear expression is an algebraic expression which is composed of constants, variables, and coefficients that are related by only addition and multiplication operations. It usually takes the form of ax + b, where 'a' and 'b' are constants and 'x' is a variable. Linear expressions can be used to describe various mathematical and physical phenomena, such as the slope of a line or the volume of a cylinder.

This linear expression is not equivalent to the other three because it does not have the same coefficient for the x terms and the constant terms. The other three expressions all have the same coefficients for the x terms and the constant terms, making them equivalent.

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The equation of the line that passes through (7, 6) and (-1, 2) in slope-intercept form is, [tex]y=\frac{1}{2}x + \frac{5}{2}[/tex].

What is slope intercept form of the line?

Using a straight line's slope and the location of its y-axis intersection, the slope intercept equation can be used to determine the general equation of the line. y = mx + b is the equation in slope intercept form.

Given:

The line passes through the points (7, 6) and (- 1, 2).

We have to find the equation of the line that passes through (7, 6) and (-- 1, 2) in slope-intercept form.

First to find the slope of the line using the formula,

Slope = m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

Here, [tex](x_1, y_1)= (7, 6), (x_2,y_2)=(-1, 2)[/tex]

Plug the values in the above formula.

[tex]m = \frac{2-6}{-1-7} = \frac{-4}{-8}=\frac{1}{2}[/tex]

Now consider the point slope form of the line,

[tex]y-y_1=m(x-x_1)\\y-6=\frac{1}{2}(x-7) \\y-6=\frac{1}{2}x-\frac{7}{2}\\ y=\frac{1}{2}x -\frac{7}{2}+6 \\y=\frac{1}{2}x+\frac{5}{2}[/tex]

Hence, the equation of the line that passes through (7, 6) and (-1, 2) in slope-intercept form is, [tex]y=\frac{1}{2}x + \frac{5}{2}[/tex].

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The rate exists water being poured into the cup when the water level is 9 cm exists [tex]$$1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

To determine the derivative of a composite function in differential calculus, apply the chain rule formula. If y = f(g(x)), then according to the chain rule, the instantaneous rates of change of the functions f and g with respect to x produce the instantaneous rate of change of f with respect to x.

A composition of functions, such as the composition of the functions f and g, f(g(x)), can be used to determine the derivative using the chain rule.

Let us set up the following variables:

R, Radius of conical cup (cm) = 30 cm

H, Height of the conical cup (cm) = 10 cm

Our aim is to find dV/dt when h = 9 and we are given dh/dt = 2

By similar triangles we have

r : h = R : H

r/h = 30/10

⇒ r = 3h

The volume of a cone is 1/3π r²h, So:

V = 1/3π r²h

simplifying the equation, we get

= 1/3π (3h²)h

= 1/3π 9h²h

= 3π h³

Differentiating wrt h we get:

dV/dh = 9π h²

Applying the chain rule we have:

dV/dt = (dV/dh)(dh/dt)

= 9π h² × 2

= 18π h²

If h = 9 then

[tex]$\left[\frac{d V}{d t}\right]_{h=9}=18 \pi\left(9^2\right)=1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

Therefore, the rate exists water being poured into the cup when the water level 9 cm exists [tex]$$1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

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Answer:

1/ 0.8 kilograms

2/ 800 grams

Step-by-step explanation:

We know

1/

Baseball bat = 28 ounces

1 kilogram = 35 ounces

To find weight in kg we take

28 / 35 = 0.8 kg

2/

We know

1 kg = 1000 g

0.8 kg = 800 grams

So, the baseball bat weights 0.8kg or 800 grams

Example of an undecidable problem is given by the problems in the system created by computer viruses.

According to the computability theory, the problems that needs either yes or no answer, but it is not possible by any computer programming languages to create exact answer then this computational problem is termed as undecidable problems.

A decidable problem is a decision problem in the computer system that can be solved by an algorithm correctly.

whenever we find a problem for which there is no algorithm available for solve and we are also unable to create any program that can solve the problem exactly in finite time are declared as undecidable problems.

If we find a problem that ask yes or no question about some input and there is programming language available for responding correctly then this particular problem is declared as decidable.

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The piecewise function represented by the graph is:

f(x) = 2x when -3 ≤ x < 0

f(x) = 3 When 1/2 < x < 3/2

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

Let f(x) = y = mx + c

From the graph, when -3 ≤ x < 0.

(-3, -6) and (-1, -2)

m = (-2 + 6) / (-1 + 3) = 4 / 2 = 2

(-1, -2) = (x, y)

-2 = 2 x (-1) + c

-2 = -2 + c

c = 0

f(x) = 2x

From the graph, When 1/2 < x < 3/2.

It is a horizontal line of y = 3.

f(x) = 3

Thus,

f(x) = 2x when -3 ≤ x < 0

f(x) = 3 When 1/2 < x < 3/2

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The two equation to set up to solve the equation |3x - 2| = 19 are

Question 2

x = 5 OR -9

Question 3

x ≥ 6 OR x ≥ -9

Without taking direction into account, absolute value describes how far away from zero a certain number is on the number line.

A number can never have a negative absolute value.

The equation is written in the form below

|3x - 2| = 19

removing the absolute value sign

3x - 2 = ± 19

the two equations are

3x - 2 = 19

3x - 2 = -19

Question 2

solving for x

3|x + 2| - 7 = 14

3|x + 2| = 14 + 7

3|x + 2| =  21

divide through by 3

|x + 2| =  7

removing the absolute value sign

x + 2 =  ±7

solving for x for positive value

x + 2 = 7

x = 5

solving for the negative sign

x + 2 = -7

x = -9

Question 3

-2|2x+3| + 14 ≥ -16

-2|2x+3| ≥ -30

dividing through by -2

|2x+3| ≥ 15

2x+3 ≥ ±15

solving for the positive sign

2x+3 ≥ 15

x ≥ 6

solving for the negative sign

2x+3 ≤ -15

x ≥ -9

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Answer: The function you provided describes an experiment in which a coin is tossed, a number cube is rolled, and a letter is picked from the word "framer". The outcome of the experiment is represented by the ordered triple (tails, 5, m).

P(tails, 5, m) refers to the probability of the outcome (tails, 5, m) occurring. To find this probability, you would need to know the probability of each individual event:

The probability of getting tails is 1/2,

The probability of getting a 5 on a number cube is 1/6

and probability of getting letter "m" in the word "framer" is 1/6

So P(tails, 5, m) = (1/2) * (1/6) * (1/6) = 1/72

It is to be noted that in order for the inverse function to exist, the function must be one-to-one, meaning each y value has one unique x value, meaning the random variables in question should be independent. But in this case the events are dependent on one another.

Step-by-step explanation:

Answer:

2800

Step-by-step explanation:

20% of 14,000 is 2800

Multiply both sides by 14,000, then divide the left side to get the result.

Answer:

2800

is what the calculator says

The price of the sales tax on the elephant could be 0.75$

What is percentage ?

percentage can be defined as the product of ratio of given value and total value and hundred.

Given

he sells a stuffed elephant for $14.95,

the sales tax is 5% of the sale price .

So,

as the tax is 5% of the given value.

so we are here multiplying the given value to the 5/100

we get,

The amount of the tax could be

 = 14.95*5/100

 = 0.7475

≈ 0.75 (rounded)

Hence the  price of the sales tax on the elephant could be 0.75$

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The value of P(X<Y) could be p(1−q) / p+q−pq if  X ∼ Geom(p) and Y ∼ Geom(r) are independent

What is probability ?

Probability could be defined as the ratio of number of favourable outcomes and total number of outcomes.

Given ,

X ∼ Geom(p) and Y ∼ Geom(r) are independent.

P(X<Y)=P(⋃1≤k<n,n∈N{X=k,Y=n})

=∑1≤k<n,n∈NP(X=k,Y=n)

=∑n=1∞∑k=1n−1P(X=k)P(Y=n)

=∑n=1∞∑k=1n−1p(1−p)k−1q(1−q)n−1

=pq∑n=1∞(1−q)n−11−(1−p)n−1p

=q∑n=1∞(1−q)n−1(1−(1−p)n−1)

=1−q / ∑n=1∞[(1−q)(1−p)]n−1

=1−q / 1−(1−p)(1−q)

=1−(1−p)(1−q)−q / 1−(1−p)(1−q)

=p(1−q) / p+q−pq.

Hence, The value of P(X<Y) could be p(1−q) / p+q−pq if  X ∼ Geom(p) and Y ∼ Geom(r) are independent.

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The graph of the linear function f(x) = 2/(x + 5) is attached.

The general equation of a straight line is -

y = mx + c

{m} - slope.

{c} - intercept along the y - axis.

Given is the linear function as -

f(x) = 5/(2x + 5)

Refer to the graph of the linear function attached. This graph represents the function f(x).

Therefore, the graph of the linear function f(x) = 2/(x + 5) is attached.

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Answer: −2<x<0

Step-by-step explanation: 2<x+4<4 2+−4<x+4+−4<4+−4 −2<x<0

Step-by-step explanation:

what do you not understand ? please let me know where you need more instructions.

the basic rule : every term of one side has to be multiplied with every term of the other side, and the results are added (by using the correct signs of the products).

remember :

+ × + = +

- × - = +

+ × - = -

- × + = -

(3x - 3)(5x - 4) = 3x × 5x + 3x × -4 + -3 × 5x + -3 × -4 =

= 15x² - 12x - 15x + 12 = 15x² - 27x + 12

It will take Emily 12.1 hours to mow 5 acres

Variation is the relation between a set of values of one variable and a set of values of other variables

Let T and A represent the time and area respectively

Since the time it takes Emily to mow a lawn is proportional to its area. We can write:

T α A

T = kA (where k is constant of proportionality)

It takes her about 30 minutes to mow 1,000 square yards. That is:

T = 30 and A = 1,000

T = kA

30 = 1000k

k = 30/1000

k = 0.03

Time for 5 acres:

5 acres = 5 × 4840 = 24200 square yards   (1 acre = 4840 square yards)

T = kA

T = 0.03 × 24200

T = 726 minutes     (divide by 60 to get time in hours)

T = 12.1 hours

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[tex](69)(5.6\times 10^{5})\implies (69)(560000)\implies 3\underset{\leftarrow 7\to }{\underline{8640000}}\implies 3.864\times 10^{7}[/tex]

The solution of given triangle is 32°.

Therefore, an isosceles triangle has two equal sides as well as two equal angles. The Greek words iso (same) and skelos are the source of the name (leg). An equilateral triangle is one in which all of its sides are equal, and a scalene triangle is one in which none of its sides are equal.

Given,

∠BAC=52°

since, ΔBAC is isosceles triangle hence opposite angle will be equal.

∠BAC+∠ABC+∠ACB=180°

2∠BCA=180°-52°

∠BCA=128/2

∠BCA=64°

also,

∠BCA+∠DCA=180°

∠ACD=180°-64°

∠ACD=116°

since,

ΔACD is isosceles triangle, hence opposite angle is equal

∠ACD+∠CAD∠ADC=180°

2∠CAD=180°-116°

∠CAD=64/2

∠CAD=32°

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