Exponents Properties Practice

Write an equation to model the situation and answer the question. Include units when applicable.

In a much happier economy, Mr. Demo earns 5% monthly interest on his savings. After a $300 withdrawal, he notices he has $2021 in his account. He has collected interest for 3 months. What amount did he start with?

Answer:

The function [tex]f:\mathbb Z\to\mathbb Z,~f(x)=3x+4[/tex] is injective (one-to-one).

Step-by-step explanation:

The definition of an injective function follows.

Let [tex]X,Y[/tex] be sets. Let [tex]f:X\to Y[/tex] be a function. We say [tex]f[/tex] is injective if, for all [tex]x,y\in X[/tex], [tex]f(x)=f(y)[/tex] implies [tex]x=y[/tex].

This is the proof that  [tex]f:\mathbb Z\to\mathbb Z,~f(x)=3x+4[/tex] is injective.

Let [tex]x,y\in\mathbb Z[/tex] and assume [tex]f(x)=f(y)[/tex]. This means [tex]3x+4=3y+4[/tex]. Subtracting [tex]4[/tex] gives [tex]3x=3y[/tex], then dividing by [tex]3[/tex] gives [tex]x=y[/tex]. Thus [tex]f[/tex] is injective.

[tex]\\ \sf\longmapsto \dfrac{1-cos2A}{1-sin2A}[/tex]

[tex]\boxed{\sf \dfrac{cosA}{sinA}=cotA}[/tex]

[tex]\\ \sf\longmapsto \dfrac{1-cos2A}{1-sin2A}[/tex]

[tex]\\ \sf\longmapsto 1-cot2A[/tex]

[tex]\\ \sf\longmapsto 1-\dfrac{1}{tan2A}[/tex]

[tex]\\ \sf\longmapsto \dfrac{tan2A-1}{tan2A}[/tex]

[tex]\\ \sf\longmapsto tan2A[/tex]

Answer:

0.8997 = 89.97% probability of a bulb lasting for at most 622 hours.

Step-by-step explanation:

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Mean of 590 hours, standard deviation of 25 hours.

This means that [tex]\mu = 590, \sigma = 25[/tex]

Find the probability of a bulb lasting for at most 622 hours.

This is the p-value of Z when X = 622.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{622 - 590}{25}[/tex]

[tex]Z = 1.28[/tex]

[tex]Z = 1.28[/tex] has a p-value of 0.8997.

0.8997 = 89.97% probability of a bulb lasting for at most 622 hours.

Answer: LCM = 1320

Step-by-step explanation:

2 | 220, 440, 660

2 | 110, 220, 330

2 | 55, 110, 165

3 | 55, 55,165

5 | 55, 55 , 55

11 | 11, 11, 11

| 1, 1, 1

= 2 × 2 × 2 × 3 × 5 × 11

= 1320

Therefore the LCM is 1320

Must click thanks and mark brainliest

Answer:

16x⁴+16x³-12x²-32x-16

Step-by-step explanation:

(8x²-4x-8)(2x²+3x+2)

= 16x⁴+24x³+16x²-8x³-12x²-8x-16x²-24x-16

= 16x⁴+16x³-12x²-32x-16

The correct option is A because

The statement makes sense. There is​ 95% probability that the confidence interval limits actually contain the true value of the population​ mean, so the probability it does not fall in this range is ​100%−​95% =​5%.

From the question we are told that:

Confidence interval [tex]CI=95\%[/tex]

Mean [tex]\=x =1.9-3.5hours[/tex]

Level of significance (of the alternative hypothesis)

[tex]\alpha=100-95[/tex]

[tex]\alpha=5\%[/tex]

[tex]\alpha=0.05[/tex]

Generally

There is​ 95% probability that the confidence interval limits actually contain the true value of the population​ mean.

In conclusion

The  it does not fall in this range is Level of significance (of the alternative hypothesis)

​100%−​95% =​5%.

For more information on this visit

https://brainly.com/question/24131141?referrer=searchResults

In this question, we have to identify the zeros of the polynomial, along with a point, and then we get that the formula for the polynomial is:

[tex]p(x) = -0.5(x^3 - x^2 + 6x)[/tex]

------------------------

Equation of a polynomial, according to it's zeros:

Given a polynomial f(x), this polynomial has roots such that it can be written as: , in which a is the leading coefficient.

------------------------

Identifying the zeros:

Given the graph, the zeros are the points where the graph crosses the x-axis. In this question, they are:

[tex]x_1 = -2, x_2 = 0, x_3 = 3[/tex]

Thus

[tex]p(x) = a(x - x_{1})(x - x_{2})(x-x_3)[/tex]

[tex]p(x) = a(x - (-2))(x - 0)(x-3)[/tex]

[tex]p(x) = ax(x+2)(x-3)[/tex]

[tex]p(x) = ax(x^2 - x + 6)[/tex]

[tex]p(x) = a(x^3 - x^2 + 6x)[/tex]

------------------------

Leading coefficient:

Passes through point (2,-8), that is, when [tex]x = 2, y = -8[/tex], which is used to find a. So

[tex]p(x) = a(x^3 - x^2 + 6x)[/tex]

[tex]-8 = a(2^3 - 2^2 + 6*2)[/tex]

[tex]16a = -8[/tex]

[tex]a = -\frac{8}{16} = -0.5[/tex]

------------------------

Considering the zeros and the leading coefficient, the formula is:

[tex]p(x) = -0.5(x^3 - x^2 + 6x)[/tex]

A similar problem is found at https://brainly.com/question/16078990

The formula that represents the polynomial in the figure is [tex]p(x) = x^{3}-x^{2}-6\cdot x[/tex].

Based on the Fundamental Theorem of Algebra, we understand that Polynomials with real Coefficient have at least one real Root and at most a number of Roots equal to its Grade. The Grade is the maximum exponent that Polynomial has and root is a point such that [tex]p(x) = 0[/tex]. By Algebra we understand that polynomial can be represented in this manner known as Factorized form:

[tex]p(x) = \Pi\limits_{i=0}^{n} (x-r_i)[/tex] (1)

Where:

[tex]n[/tex] - Grade of the polynomial.

[tex]i[/tex] - Index of the root binomial.

[tex]x[/tex] - Independent variable.

We notice that polynomials has three roots in [tex]x = -2[/tex], [tex]x = 0[/tex] and [tex]x = 3[/tex], having the following construction:

[tex]p(x) =(x+2)\cdot x \cdot (x-3)[/tex]

[tex]p(x) = (x^{2}+2\cdot x)\cdot (x-3)[/tex]

[tex]p(x) = x^{3}+2\cdot x^{2}-3\cdot x^{2}-6\cdot x[/tex]

[tex]p(x) = x^{3}-x^{2}-6\cdot x[/tex]

The formula that represents the polynomial in the figure is [tex]p(x) = x^{3}-x^{2}-6\cdot x[/tex].

Here is a question related to the determination polynomials: https://brainly.com/question/10241002

⇛6(2 - 6x) = -24

⇛12 - 36x = -24

⇛-36x = -24 - 12

⇛-36x = -36

⇛x = -36/-36

⇛x = 1

Answer:

c=80

Step-by-step explanation:

Based on my reading the critical damping occurs when the discriminant of the quadratic characteristic equation is 0.

So let's see that characteristic equation:

20⁢r^2+c⁢r+80⁢=0

The discriminant can be found by calculating b^2-4aC of ar^2+br+C=0.

a=20

b=c

C=80

c^2-4(20)(80)

We want this to be 0.

c^2-4(20)(80)=0

Simplify:

c^2-6400=0

Add 6400 on both sides:

c^2=6400

Take square root of both sides:

c=80 or c=-80

Based on further reading damping equations in form

a⁢y′⁢′+b⁢y′+C⁢y=0

should have positive coefficients with b also having the possibility of being zero.

3385.8

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

You should find the solution for this system of equations (The value of x is the first coordinate for the point of intersection, the value of y is the second coordinate for the point of intersection)

X+y=0.3

y=3x+16.7

use 3x+16.7 instead of y in the first equation(do it to get only x in the first equation)

x+3x+16.7=0.3

4x+16.4=0

4x=-16.4

x=-4.1

y=0.3+4.1=4.4

B

Answer:

It is (x - 3)³ - 9x(3 - x)

Step-by-step explanation:

Express 27 in terms of cubes, 27 = 3³:

[tex] = {x}^{3} - {3}^{3} [/tex]

From trinomial expansion:

[tex] {(x - y)}^{3} = (x - y)(x - y)(x - y) \\ [/tex]

open first two brackets to get a quadratic equation:

[tex] {(x - y)}^{3} = ( {x}^{2} - 2xy + {y}^{2} )(x - y)[/tex]

expand further:

[tex] {(x - y)}^{3} = {x}^{3} - y {x}^{2} - 2y {x}^{2} + 2x {y}^{2} + x {y}^{2} - {y}^{3} \\ {(x - y)}^{3} = {x}^{3} - {y}^{3} + 3x {y}^{2} - 3y {x}^{2} \\ {(x - y)}^{3} = {x}^{3} - {y}^{3} + 3xy(y - x) \\ \\ { \boxed{( {x}^{3} - {y}^{3} ) = {(x - y)}^{3} - 3xy(y - x)}}[/tex]

take y to be 3, then substitute:

[tex]( {x}^{3} - 3^3) = {(x - 3)}^{3} - 9x(3 - x)[/tex]

Answer:

432 inches

Step-by-step explanation:

We need to convert feet to inches

1 ft = 12 inches

36 ft * 12 inches/ 1 ft = 432 inches

9514 1404 393

Answer:

  x = 1/7

Step-by-step explanation:

  x³ = 1/343

  x = 1/∛343 = 1/7

The positive value of x is 1/7.

_____

Additional comment

A square root (or any even-index root) will have both positive and negative real values. A cube root (or any odd-index root) will have only one real value, whose sign will match the sign of the value being rooted.

343^(-1/3) is the cube root of a positive number, so it will be a positive real number.

Answer:  [tex]-\frac{\sqrt{2a}}{8a}[/tex]

=======================================================

Explanation:

The (x-a) in the denominator causes a problem if we tried to simply directly substitute in x = a. This is because we get a division by zero error.

The trick often used for problems like this is to rationalize the numerator as shown in the steps below.

[tex]\displaystyle \lim_{x\to a} \frac{\sqrt{3a-x}-\sqrt{x+a}}{4(x-a)}\\\\\\\lim_{x\to a} \frac{(\sqrt{3a-x}-\sqrt{x+a})(\sqrt{3a-x}+\sqrt{x+a})}{4(x-a)(\sqrt{3a-x}+\sqrt{x+a})}\\\\\\\lim_{x\to a} \frac{(\sqrt{3a-x})^2-(\sqrt{x+a})^2}{4(x-a)(\sqrt{3a-x}+\sqrt{x+a})}\\\\\\\lim_{x\to a} \frac{3a-x-(x+a)}{4(x-a)(\sqrt{3a-x}+\sqrt{x+a})}\\\\\\\lim_{x\to a} \frac{3a-x-x-a}{4(x-a)(\sqrt{3a-x}+\sqrt{x+a})}\\\\\\[/tex]

[tex]\displaystyle \lim_{x\to a} \frac{2a-2x}{4(x-a)(\sqrt{3a-x}+\sqrt{x+a})}\\\\\\\lim_{x\to a} \frac{-2(-a+x)}{4(x-a)(\sqrt{3a-x}+\sqrt{x+a})}\\\\\\\lim_{x\to a} \frac{-2(x-a)}{4(x-a)(\sqrt{3a-x}+\sqrt{x+a})}\\\\\\\lim_{x\to a} \frac{-2}{4(\sqrt{3a-x}+\sqrt{x+a})}\\\\\\[/tex]

At this point, the (x-a) in the denominator has been canceled out. We can now plug in x = a to see what happens

[tex]\displaystyle L = \lim_{x\to a} \frac{-2}{4(\sqrt{3a-x}+\sqrt{x+a})}\\\\\\L = \frac{-2}{4(\sqrt{3a-a}+\sqrt{a+a})}\\\\\\L = \frac{-2}{4(\sqrt{2a}+\sqrt{2a})}\\\\\\L = \frac{-2}{4(2\sqrt{2a})}\\\\\\L = \frac{-2}{8\sqrt{2a}}\\\\\\L = \frac{-1}{4\sqrt{2a}}\\\\\\L = \frac{-1*\sqrt{2a}}{4\sqrt{2a}*\sqrt{2a}}\\\\\\L = \frac{-\sqrt{2a}}{4\sqrt{2a*2a}}\\\\\\L = \frac{-\sqrt{2a}}{4\sqrt{(2a)^2}}\\\\\\L = \frac{-\sqrt{2a}}{4*2a}\\\\\\L = -\frac{\sqrt{2a}}{8a}\\\\\\[/tex]

There's not much else to say from here since we don't know the value of 'a'. So we can stop here.

Therefore,

[tex]\displaystyle \lim_{x\to a} \frac{\sqrt{3a-x}-\sqrt{x+a}}{4(x-a)} = -\frac{\sqrt{2a}}{8a}\\\\\\[/tex]

9514 1404 393

Answer:

Step-by-step explanation:

The coefficient of a term is its constant multiplier. The exponent is the power to which the base is raised.

The term 4·b^(-3) has an exponent of -3, a base of b and a coefficient of 4.

The exponent is -3; the coefficient is 4.

Answer:

exponent = -3           coefficent = 4

Step-by-step explanation:

[tex]\\ \sf\longmapsto Area=Length\times Breadth[/tex]

[tex]\\ \sf\longmapsto Area=48(36)[/tex]

[tex]\\ \sf\longmapsto Area=1728in^2[/tex]

[tex]\\ \sf\longmapsto Area=144ft^2[/tex]

[tex]\\ \sf\longmapsto Area=48yard^2[/tex]

Answer:

(3,1) is the midpoint

Step-by-step explanation:

To find the x coordinate of the midpoint, average the x coordinates of the endpoints

(7+-1)/2 = 6/2 =3

To find the y coordinate of the midpoint, average the y coordinates of the endpoints

(10+-8)/2 = 2/2 = 1

(3,1) is the midpoint

Answer:

(3, 1)

Step-by-step explanation:

We can use the formula [ (x1+x2)/2, (y1+y2/2) ] to solve for the midpoint.

7+(-1)/2, 10+(-8)/2

6/2, 2/2

3, 1

Best of Luck!

Answer:

D

Step-by-step explanation:

More than 40 miles per gallon

Open circle at 40 and line goes to the left

Answer: the answer is C

Step-by-step explanation:

because it is a open circle going to the left

Answer:

multiples of 2 2,4,6,8,10,12,14,16,18,20

multiples of 3 3,6,9,12,15,,18,21,24,27,30

Step-by-step explanation:

Lcm is 6

From the information given in the exercise, we build the confidence interval and solve this question. First, we have to find the point estimate for the population proportion, then using this point estimate, and sample size, we build the confidence interval. According to the built confidence interval, question c is answered.

Item a:

522 out of 1005 indicated that television is a luxury that they could do without, so:

[tex]\pi = \frac{522}{1005} = 0.5194[/tex]

Thus, the point estimate for the population proportion of adults in the country who believe that televisions are a luxury they could do without is 0.5194.

Item b:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the z-score that has a p-value of .

For this problem, we have that:

[tex]n = 1005,\pi = 0.5194[/tex]

95% confidence level

So [tex]\alpha = 0.05[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].  

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.5194 - 1.96\sqrt{\frac{0.5194*0.4806}{1005}} = 0.4885[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.5194 + 1.96\sqrt{\frac{0.5194*0.4806}{1005}} = 0.5503[/tex]

Thus, the 95% confidence interval for the population proportion of adults in the country who believe that televisions are a luxury they could do without is (0.4885,0.5503). The interpretation is that:

We are  95% confident the proportion of adults in the country who believe that televisions are a luxury they could do without is between 0.4885 and 0.5503.

Item c:

It is possible, but unlikely that a supermajority of adults in the country believe that television is a luxury they could do without because the 95% confidence  interval does not contain 60%.

For another example of a confidence interval for a proportion, you can check https://brainly.com/question/16807970

Answer:

0.5*sqrt33

Step-by-step explanation:

A(0,0,0) B(-4,1,-2), c(-4,2,-3)

Vector AB is (-4-0,-1-0, -2-0)= (-4,-1,-2)  The modul of AB is sqrt (4squared+

+(-1) squared+ (-2) squared)= sqrt (16+1+4)=sqrt21

Vector AC is (-4,2,-3) The modul of vector AC is equal to sqrt ((-4)squared+ 2squared+(-3)squared)= sqrt(16+4+9)= sqrt29

Vector BC is equal to (-4-(-4), 2-1, -3-(-2))= (0,1,-1)

The modul of BC is sqrt (1^2+(-1)^2)=sqrt2

Find the angle B

Ac^2= BC^2+AB^2-2*BC*AB*cosB

29= 2+21-2*sqrt2*sqrt21*cosB

29= 2+21-2*sqrt42*cosB

cosB= -3/ sqrt42

sinB= sqrt( 1-(-3/sqrt42)^2)=sqrt33/42= sqrt11/14

s=1/2* (sqrt2*sqrt21*sqrt11/14)=1/2*sqrt(42*11/14)= 0.5*sqrt33

Answer:

The leading coefficient is -8 as it is a mix of x and cardinal, if it was x alone then it wouldn't be the coefficient, we would use the next number shown.

If it was just a number and no x then it would still be the coefficient.

The degree is 9 as it is the highest power shown.

Step-by-step explanation:

See attachment for examples

Answer:

16.25

Step-by-step explanation:

first do 12 -5 = 7. then 7/4 = 1.75 then 9+1.75 = 10.75 and finally 27-10.75= 16.25

Speeding up velocity for 5 seconds, same speed for another 10 seconds, slows down for 10 seconds.

Answer:

Variance is 256

Step-by-step explanation:

Variance:

[tex]var = \frac{ ({ \sum x})^{2} }{n} - {( \frac{ \sum x}{n} })^{2} [/tex]

x is the number or item in the data

n is the number of terms

[tex]{ \tt{ \sum x = (12 + 7 + 6 + 4 + 11)}} \\ { \tt{ \sum x = 40}}[/tex]

Therefore:

[tex]variance = \frac{ {40}^{2} }{5} - { (\frac{40}{5}) }^{2} \\ \\ = 320 - 64 \\ variance = 256[/tex]

Answer:

2 years

Step-by-step explanation:

population in 1 year= 24400*105%=25620

population in 2 year= 25620*105%=26901

Step-by-step explanation:

Here are some of the graphs:

Blue is g(x) and Green is f(x). The 2nd graph is for the 13b. It shows our graph after 1 transformation. The 3rd graph is after both transformations.

13a. Let use the following values in

[tex]f(x) = \frac{2}{x} [/tex]

We know by definition of rational function x cannot be zero.

Let find some values across interval 2 through 4.

[tex]f(2) = \frac{2}{2} = 1[/tex]

[tex]f(3) = \frac{2}{3} [/tex]

[tex]f(4) = \frac{2}{4} = \frac{1}{2} [/tex]

Let use the following values in

[tex]g(x) = \frac{3x - 1}{x - 1} [/tex]

By definition of rational function, x cannot be 1 because it will make the denominator zero. Let use some values across the interval 0 through 4.

[tex]g(0) = \frac{0 - 1}{0 - 1} = 1[/tex]

[tex]g(2) = \frac{3(2) - 1}{2 - 1} = {5} [/tex]

[tex]g(3) = \frac{8}{2} = 4[/tex]

[tex]g(4) = \frac{11}{3} [/tex]

So graph this in a table of values. I'll post a picture of the table of values on the top.

13b. We need to write g(x) as a transformation of f(x). If we look at the graphs, g(x) has a asymptote at x=1 while f(x) has a asymptote of 0. This means that we need to move f(x) to the right one unit or move (x-1) units.

We will upgrade the graph.

Now we can just add 3 to f(x) to get to g(x).

In the 3rd graph, notice how both graphs coincide. Our transformations is complete.

The answer is

[tex]g(x) = f(x - 1) + 3[/tex]

13c. We can say this as we move f(x) to the right 1 unit and shift f(x) up 3 units.