How is i 4 equal to 1?
The given expression is a complex expression and we can prove the given expression equal to 1 by using definition.
What is i in complex number?a+ib, often known as the generic form of complex numbers, is how complex numbers are expressed. In the complex numbers a+ib form, a represents the real component of the number, b represents the imaginary part, and i is defined as√(-1). Complex numbers can take a variety of shapes.
Write the general form of complex number.z=a+bi is the general form of complex number
As by definition
i^2=-1
So
i^4=(i^2)^2
=(-1)^2
=1
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slope=8 y-intercept=2
Answer:
y=8x+2.
Step-by-step explanation:
Slope intercept equation: y=mx + or - b.
m or slope is 8. plug that into the slope intercept equation of y=mx+b
2 is positive. so its +2 after 8x.
A model airplane in Yesinia's collection is at a 1:40 scale to the actual plane. If the model is 7.5 inches long, what are the dimensions of the actual airplane in feet?
The dimension of actual plane is 300 inches.
What are ratios?
A ratio is the comparison of quantities which have same parameters or units of measurement.
Why are calculation of ratios significant?
It helps to identify how big or small a quantity is as compared to other thing having same parameter of measurement.
NOTE: Things which have different measurement cannot be compared by ratio;
Example: 1 litre of milk cannot be in ratio with 1 kg of sand, as milk is measured in volume and sand is in weight.
Solution:
Given, ratio of model of plane and actual plane = 1 : 40
x = 7.5 inches
Let, x be the dimension of model of the airplane
y be the dimension of actual airplane
Using the formula by equating the ratios,
1/40 = x/y
⇒ 1 x y = 40x
⇒ y = 40 x 7.5
⇒ y = 300 inches
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anja simplified the expression startfraction 10 x superscript negative 5 baseline over negative 5 x superscript 10 baseline endfraction to startfraction 15 over x superscript 15 baseline endfraction. what mistake did anja make?
The simplified form of the expression 10x⁻⁵/-5x¹⁰ is -2/x¹⁵. So Anja wrote the numerator as 15 instead of -2.
What is Exponentiation?Exponentiation is the method of writing large numbers as the number in terms of power.
Exponent indicates how much time a number is multiplied to itself.
aᵇ read as a raise to b, implies that a is multiplies to itself b times.
We have to simplify the expression 10x⁻⁵/-5x¹⁰.
[tex]\frac{10x^-5}{-5x^1^0}[/tex] = 10 × [tex]\frac{1}{-5}[/tex] × x⁻⁵ × x⁻¹⁰
= -2 x⁻¹⁵
= [tex]\frac{-2}{x^1^5}[/tex]
Hence Anja simplified the expression as [tex]\frac{15}{x^1^5}[/tex] which is not true. The correct simplified form is [tex]\frac{-2}{x^1^5}[/tex].
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Answer:
its B
Step-by-step explanation:
i did the test
Determine whether the lines are parallel, perpendicular, or neither
X+7y = -3 and y= 7x + 25
Answer:
perpendicular
Step-by-step explanation:
7y = -x - 3
y = -1/7 x - 3/7
-1/7 is the reciprocal of the opposite, so the lines are perpendicular
What is the difference between continuous function and discontinuous function?
A continuous function is a function that is defined for all values of the independent variable, and its graph has no breaks or jumps. A discontinuous function is a function that is not continuous, meaning there are values of the independent variable at which the function is undefined or has a break or jump in its graph.
A continuous function is a function that can be graphed without any breaks or jumps. This means that the graph of a continuous function is a single, unbroken curve that passes through all its points. Furthermore, a continuous function is defined for all values of the independent variable. This means that the function is defined for all points in its domain. On the other hand, a discontinuous function is a function that is not continuous, meaning there are values of the independent variable at which the function is undefined or has a break or jump in its graph. This means that the graph of a discontinuous function will have jumps or breaks. Furthermore, a discontinuous function is not defined for all values of the independent variable, meaning that there are certain points in the domain for which it is not defined. In conclusion, the main difference between continuous and discontinuous functions is that a continuous function is defined for all values of the independent variable and has no jumps or breaks in its graph, whereas a discontinuous function is not defined for all values of the independent variable and has jumps or breaks in its graph.
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Why can't the square root of 2 be written as a fraction?
The square root of 2 is "irrational" (cannot be represented as a fraction) because, if it could, we would have the nonsensical circumstance .
what is square root ?A square root is a factor of a number in mathematics that, when multiplied by itself, equals the original value. The square roots of 9, for instance, are 3 and -3. The Babylonians developed efficient approaches to approximate square roots as early as the second millennium BC.
here,
The square root of 2 is "irrational" (cannot be represented as a fraction) because, if it could, we would have the nonsensical circumstance that the fraction would have even integers at both the top and bottom and could thus always be simplified.
As a result, the square root cannot be expressed as a fraction.
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12 The acute angle between the vectors ( 1 k 2 ) and ( 0 -1 1 ) is 45° Calculate the possible values of k
the answer is -1/4 but i have no clue how to reach it, thanks.
The confidence interval of p±E is 0.555±0.444
Definition of confidence interval-A Confidence Interval is a region constructed using sampled data, of fixed size, from a population (sample space) following a certain probability distribution. The interval is constructed to contain a chosen population statistic with prescribed probability.
A confidence interval, in statistics, refers to the probability that a population parameter will fall between a set of values for a certain proportion of times. Analysts often use confidence intervals than contain either 95% or 99% of expected observations. Thus, if a point estimate is generated from a statistical model of 10.00 with a 95% confidence interval of 9.50 - 10.50, it can be inferred that there is a 95% probability that the true value falls within that range.
Given:-0.111<p<0.999=Confidence interval. we have to express it in the form p±E.
Explanation:-The midpoint of the interval is p.
p=[tex]\frac{0.111+0.999}{2}[/tex]=0.555
Next the error term E is the distance from p to the end of the interval.
E=0.999-0.555=0.444
So, finally we get the answer as p±E=0.555±0.444
The final answer is p±E=0.555±0.444
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PNQ has vertices P(2,5), N(-3,-1), and Q(4,0)
The following transformation will be applied: (x, y) —> (x + 5, y + 6)
A. Write the translation in words:
________________________
For each point, determine the new image point.
P (2,5) ——> P’ (______)
N (-3,-1) ——> N’ (______)
Q (4,0) ——> Q’ (______)
WILL GIVE BRAINLIEST IF CORRECT!!
Answer:
A. Triangle PNQ is translated along the vector <5,6> to map onto triangle P'N'Q' OR
A. Triangle PNQ is translated right 5 units and up 6 units to map onto P'N'Q'
P (2,5) --> P' (7,11)
N (-3,-1) --> N' (2,5)
Q (4,0) --> Q' (9,6)
Step-by-step explanation:
What are the 5 types of polynomial?
Answer: Monomial, binomial,trimonial, polynomials
Step-by-step explanation:
How do you calculate decay from half-life?
Using the formula T1/2 = 0.693/λ, we may determine decay from half-life.
What is half-life?The half-life is the amount of time needed for half of the initial population of radioactive atoms to decay.
T1/2 = 0.693/λ describes the relationship between the half-life, T1/2, and the decay constant.
A percentage is used to represent the degradation rate.
Simply decreasing the percent and dividing it by 100 yields the decimal equivalent.
The decay factor b = 1-r can therefore be calculated.
For instance, the exponential function's decay rate is 0.25, and the decay factor b = 1- 0.25 = 0.75 if the rate of decay is 25%.
Nearly all decay processes that are exponential, or nearly exponential, are referred to by the phrase "half-life".
Therefore, using the formula T1/2 = 0.693/λ, we may determine decay from half-life.
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A peice of cable 8.5 cm long wieghs 52 grams. What will a 10-cm length if the same cable weigh
If the length of the cable is 10 cm. Then the weight of the cable will be 61.18 grams.
What are ratio and proportion?A proportion is a gathering of consecutively requested numbers an and b communicated as a/b, where b is never equivalent to nothing. At the point when two items are equivalent, a proclamation is supposed to corresponding.
The rate is the ratio of the amount of something to the unit. For example - If the speed of the car is 20 km/h it means the car travels 20 km in one hour.
The ratio of the weight to length will remain constant. Then the rate is calculated as,
Rate = 52 / 8.5
Rate = 6.1176 grams per cm
If the length of the cable is 10 cm. Then the weight of the cable is given as,
Weight = 6.1176 x 10
Weight = 61.18 grams
Assuming the length of the link is 10 cm. Then, at that point, the heaviness of the link will be 61.18 grams.
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The table shows the number of touchdowns Luis scored during his football games.
Touchdowns scored (X)
0
1
2
3
Number of games (f)
5
11
6
6
Create a probability distribution table. Show your work. Round your answers to two decimal places.
What is the probability of Luis scoring 2 touchdowns?
What is the probability of Luis scoring more than 1 touchdown? Show your work.
What is the estimated value of the number of touchdowns Luis scores? Show your work.
A probability is given by the division of the number of desired outcomes by the number of total outcomes.
The total number of games is given as follows:
5 + 11 + 6 + 6 = 28.
He scored 2 touchdowns in 6 games, hence the probability is of:
p = 6/28 = 3/14.
He scored more than 1 touchdown in 6 + 6 = 12 games, hence the probability is of:
p = 12/28 = 6/14 = 3/7.
The estimated value is the mean, which is the sum of all observations divided by the number of values, hence:
Mean = (0 x 5 + 1 x 11 + 2 x 6 + 3 x 6)/28 = 1.46 touchdowns.
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a flight from boston to london took only 5 hours because of a strong tailwind. The flight back to Boston against the same wind took 6 hours. What was the speed of the wind if the speed of the plane in still air is 550 miles per hour?
The velocity of the wind if the speed of the plane in still air is 550 miles per hour is of:
50 miles per hour.
How to obtain the velocity of the wind?To obtain the velocity of the wind, we need to consider the relation between velocity, distance and time, which states that velocity is distance divided by time, as follows:
v = d/t.
In which:
v is the velocity.d is the distance.t is the time.On the flight from Boston to London, the wind was helping the plane, hence the equation is given as follows:
v + vw = d/t
In which vw is the velocity of the wind.
We have that the parameters are given as follows:
v = 550, t = 5.
Hence:
550 + vw = d/5.
d = 5(550 + vw).
For the return flight, the wind was in the opposite direction of the plane, and the flight took 6 hours, hence:
550 - vw = d/6
Hence:
d = 6(550 - vw)
Hence the velocity of the wind can be calculated as follows:
5(550 + vw) = 6(550 - vw)
11vw = 550
vw = 550/11
vw = 50 miles per hour.
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Sydney pays a flat rate of $20 plus a
rate of 7 cents per minute in her phone
plan. Which function shows the cost, "C",
of Sydney's plans as a function of time
in minutes?
Answer:
The answer is C(t)=0.07t+20
Step-by-step explanation:
20 is the starting value, or y-intercept in this case. The rate of change is 7 cents, or, 0.07. If we put this in y=mx+b format, it will come out as:
C(t)=0.07t+20
What is the slope of 3 and 8?
The Slope of the line passing through the points (0,3) and (5,8) is 1 .
The Line is passing through the points (0,3) and (5,8) .
we know that the formula to calculate the slope of the line passing through the points(x1,y1) and (x2,y2) is denoted by ;
Slope(m) = ;
It is given that points through which line is passing is (0,3) and (5,8) .
substituting the values in the above formula ,
we get ;
Slope(m) = (8 - 3)/(5 - 0)
On simplifying further ,
we get ;
slope(m) = 5/5 = 1 .
Therefore , the slope of line is = "1" .
The given question is incomplete , the complete question is
What is the slope of the line passing through the points (0,3) and (5,8) ?
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How do you prove that two right triangles are similar?
We can prove that two right triangles are similar by:
Hypotenuse-Leg (HL) Theorem.
Leg-Acute (LA) Angle Theorem.
Hypotenuse-Acute (HA) Angle Theorem.
Leg-Leg (LL) Theorem.
Two triangles are congruent if they have the same three sides and exactly the same three angles. Two right triangles can be considered to be congruent, if they satisfy one of the following theorems.
Theorem 1 : Hypotenuse-Leg (HL) Theorem :
If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent.
Theorem 2 : Leg-Acute (LA) Angle Theorem :
If a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, then the two right triangles are congruent.
Theorem 3 : Hypotenuse-Acute (HA) Angle Theorem :
If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.
Theorem 4 : Leg-Leg (LL) Theorem :
If the legs of one right triangle are congruent to the legs of another right triangle, then the two right triangles are congruent.
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How do you find the base of an isosceles triangle with legs?
We can find the base of the isosceles triangle with legs using the formula:
2 × sqrt ((hypotenuse) ^ 2 - (altitude)^2) = base
Let us assume that the leg of the isosceles triangle is 'h' and one side is 'a'. Let the base be 'x'.
As there is a leg (altitude) in the isosceles triangle, then it forms a right triangle in the isosceles triangle, such that the sides form the hypotenuse.
Therefore, by using the Pythagoras' theorem,
hypotenuse ^ 2 = (altitude)^2 + (base / 2)^2
Hence, we get the base by deriving this expression from the above formula:
2 × sqrt ((hypotenuse) ^ 2 - (altitude)^2) = base
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For each graph, select all symmetries that apply. (a) (b) (C) y 6- y 4 4. 4-1 2 2- X x 76 TC -4 6 -6 -4 -2. -2 -2- -2- -4- -4- -4 -6 -6- -67 Symmetry: Symmetry: Symmetry: X-axis X-axis Ox-axis y-axis | y-axis y-axis origin origin origin none of these none of these none of these Х s ?
The symmetries of the graphs are mentioned below -
About the origin.About the {y} - axis.About the {x} - axis}What is the axis of symmetry?The axis of symmetry is a straight line that makes the shape of the object symmetrical. The axis of symmetry creates the exact reflections on each of its sides.Given are the graphs as shown in the image.
For each graph we can discuss the symmetry below -
Graph {A} -
A graph is said to be symmetric about the origin if whenever (a, b) is on the graph then so is (- a, - b) on the graph. So, the graph {A} is symmetric about the origin.Graph {B} -
The graph {B} is symmetrical about the {y} - axis.Graph {C} -
The graph {C} is symmetrical about the {x} - axis.Therefore, the symmetries of the graphs are mentioned below -
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Question 9 Which of the following distributions has a mean that varies? I. The population distribution II. The distribution of sample data III. The sampling distribution of the sample mean O ll only Il only O Tonly all three distributions I and III
The mean varies in the distribution of sample data. Therefore, statement II is correct and option a is the correct option.
A statistic sampling distribution is a type of probability distribution created by choosing several random samples from the same population, each of a preset size. One can better understand the variations in a sample statistic by using these distributions.
The population's mean, from which the sample of items was taken, is the mean of the sampling distribution. Therefore, if the population mean is μ, then the sampling distribution mean is also μ. In a population distribution, the mean does not vary from one sample to another. Also, the sample mean will always have the same mean in the sampling distribution. But the mean will vary if we take different samples in the distribution of sample data. Therefore, statements II is correct.
The complete question is -
Which of the following distributions has a mean that varies?
I. The population distribution
II. The distribution of sample data
III. The sampling distribution of the sample mean
a) ll only
b) III only
c) I only
d) all three distributions
e) II and III
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P(x)= 1/2x^2+3x-4x^3+6x^4-1 is it polynomial? If so write in standard form, and what degree, type and leading coefficient is it?
Standard form: [tex]6x^{4}-4x^{3}+\frac{1}{2} x^{2}-1[/tex]
Degree: 4
Leading coefficient: 6
What is a standard form of polynomial?A polynomial is written in the standard form when the term with the largest power of the variables comes first, followed by the subsequent terms in decreasing order of the variable power.
Detailed Explanation:Looking at the equation the highest power of the polynomial is 4;
The leading coefficient is nothing but the coefficient of the term which is having the highest power i.e.., 6;
Now arranging terms in the descending order(according to power of term) we get [tex]6x^{4}-4x^{3}+\frac{1}{2} x^{2}-1[/tex]
Standard form: [tex]6x^{4}-4x^{3}+\frac{1}{2} x^{2}-1[/tex]
Degree: 4
Leading coefficient: 6
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Solve the equation. Show your work. Express your answer as both a simplified mixed number and as a decimal.
-2.25 + 0.4 = -2.8
Answer:
-2.25 + 0.4 = -2.8
-1.85 = -2.8
-185/100 = -28/10
-37/20 = -14/5
a sponsor will donate $0.15 for every meter merissa walks in a charity event. Merissa walks 5 kilometers in the charity event. How much does the sponsor donate to merissa's walk
The sponsor will donate 750 dollars.
1 kilo is 1000 meters
5 kilos is 1 kilo x 5
1000 meters x 5 is 5000 meters
0.15 x 5000 = 750
Answer:
12345678900987654321
If ABC = DEC, DCE = 55, CDE = 65, and ABC = 2x
Similar triangles are those with the same form but different sizes. The expression's measure for the value of x is 32.5.
What are similar triangles?Triangles with the same shape but different sizes are said to be similar triangles. Squares with any side length and all equilateral triangles are examples of related objects.
In other words, if two triangles are identical, their respective sides are equal in number and their corresponding angles are congruent.
From the given diagram, since the triangle ABC is similar to DCE, hence the measure of the angle CDE is equal to ABC that is;
<CDE = <ABC
2x = 65
x = 65/2
x = 32.5
Hence the measure of the value of x from the expression is 32.5
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9 apples and 6 oranges cost $33. 18 apples and 20 oranges cost $80. how much does one apple and one orange cost?
Answer:
Apple: $2.50
Orange: $1.75
Step-by-step explanation:
Let A represent apples and Y represent Orange
We have equations
9A + 6Y = $33
18A + 20 Y = $80
Let's find out the cost of oranges first. Multiply the first equation by -2
-18A - 12Y = $-66
18A + 20 Y = $80
8Y = $14
Y = $1.75
Now let's put 1.75 back in the Y to find the cost of apple
18A + 20(1.75) = $80
18A + 35 = $80
18A = $45
A = $2.50
Answer:2 50
Step-by-step explanation:
How do I create a 4 quadrant chart in Word?
To create a 4 quadrant chart in Microsoft Word, follow the 13 steps given below.
1. Open a new Word document and click on the Insert tab.
2. Select Chart from the Illustrations group.
3. Choose the type of chart you want to create from the drop-down menu.
4. Click on the Design tab and select the Chart Layout option.
5. Select the Four Quadrant Chart option under the Layout section.
6. Enter the data for each quadrant into the chart.
7. Click on the Format tab and select the Chart Elements option.
8. Select the axes you want to use for the four quadrants from the drop-down menu.
9. Enter the labels for the four quadrants in the Axis Labels section.
10. Select the desired formatting options for the chart.
11. Click the Insert button to insert the chart into the document.
12. Adjust the position and size of the chart as desired.
13. Click the Save button to save the document.
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What are the possible rational roots of x³ X² 10x 8 0?
The Possible rational roots of the equation "x³- x² - 10x - 8 = 0" are x = -1 , x = -2 and x = 4 .
The Rational roots of the equation are defined as the values of the variable such that value of the entire equation becomes 0 .
the function is f(x) = x³- x² - 10x - 8 ,
By the Rational Roots Theorem , we know that is p/q is the root of the equation , then p is a factor of 8 and q is a factor of 1 .
the possible values of p are ±1 , ±2 , ±4 , ±8 , and
the possible values of q are ±1 .
So , the possible rational roots (p/q) are ±1 , ±2 , ±4 , ±8 .
So , on substituting the all the possible values of the rational roots ,
we get ;
f(1) = (1)³- (1)² - 10(1) - 8 = -18
f(-1) = (-1)³- (-1)² - 10(-1) - 8 = 0
f(2) = (2)³- (2)² - 10(2) - 8 = -24
f(-2) = (-2)³- (-2)² - 10(-2) - 8 = 0
f(4) = (4)³- (4)² - 10(4) - 8 = 0
f(-4) = (-4)³- (-4)² - 10(-4) - 8 = -48
f(8) = (8)³- (8)² - 10(8) - 8 = 360
f(-8) = (-8)³- (-8)² - 10(-8) - 8 = -446.
we get , the value as 0 at x = -1 , -2 and 4 .
Therefore , the possible rational roots are x = -1 , x = -2 and x = 4 .
The given question is incomplete , the complete question is
What are the possible rational roots of x³- x² - 10x - 8 = 0 ?
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find the foci of the ellipse whose major axis has endpoints $(0,0)$ and $(13,0)$ and whose minor axis has length 12.
The foci of the ellipse are (4,0) (9,0)
In order to present geometric forms in a two-dimensional plane and learn about their properties, coordinate geometry is a crucial area of mathematics.
The center of that ellipse is the midpoint of the major axis,
half way between (0,0) and (13,0) on line y=0.
The coordinates of that midpoint are the averages of the coordinates of the vertices.
The midpoint is (6.5,0) , and that is the center of the ellipse.
The distance from the center to each vertex is called the semi-major axis and represented as a .
In this case a=6.5 .
Half of the minor axis is called the semi-minor axis and is represented with b .
That is the distance from the center to each co-vertex.
In this case b= 12/2=6 .
The distance from the center to each focus is represented with c ,
and is related to a and b by a²=b²+c²
In this case 6.5²=6²+c²
Solving for c :
6.5²-6²=c²
A calculator can do that
6.5 -6
0.5*12.5=c²
c=√6.25
c=2.5 .
That is the distance from center (6.5,0) to each focus.
The foci are on the same line as the vertices, line y=0,
on either side of the center,
so the x-coordinate of one focus is 6.5-2.5=4 ,
and the x-coordinate of the other focus is 6.5+2.5=9 .
So, The foci of the ellipse are (4,0) (9,0
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The coordinates of that midpoint are the averages of the coordinates of the vertices.
The midpoint is (6.5,0) , and that is the center of the ellipse.
In this case a=6.5 .
Half of the minor axis is called the semi-minor axis and is represented with b .
That is the distance from the center to each co-vertex.
In this case b= 12/2=6 .
and is related to a and b by a²=b²+c²
In this case 6.5²=6²+c²
6.5²-6²=c²
A calculator can do that
6.5 -6
c=2.5 .
That is the distance from center (6.5,0) to each focus.
The foci are on the same line as the vertices, line y=0,
on either side of the center,
so the x-coordinate of one focus is 6.5-2.5=4 ,
and the x-coordinate of the other focus is 6.5+2.5=9 .
So, The foci of the ellipse are (4,0) (9,0)
How do you find the angle of a triangle with 3 side lengths?
By using the SSS triangle property we will find the angle of the triangle.
SSS means-Side Side Side
"SSS" is when we know three sides of the triangle, and want to find the missing angles.
To solve an SSS triangle we use the following steps:
First to calculate one of the angles we use the Law of Cosines
Apply the same process the Law of Cosines again to find another angle.
finally, use the angles of a triangle to add to 180° to find the last angle.
We use the "angle" of the Law of Cosines:
[tex]cos(C)=\frac{a^2+b^2-c^2}{2ab}[/tex]
[tex]cos(A)=\frac{b^2+c^2-a^2}{2ab}[/tex]
[tex]cos(B)=\frac{a^2+c^2-b^2}{2ab}[/tex]
By using these formulas the angles of a triangle are measured.
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Is f x )= π a polynomial function?
Yes, f(x) = π is a Polynomial function with the highest power = 0.
Let's First understand what is a polynomial function.
In an equation such as the quadratic equation, cubic equation, etc., a polynomial function is a function that only uses non-negative integer powers or only positive integer exponents of a variable. For instance, the polynomial 2x+5 has an exponent of 1.
Given function f(x) = π is also a polynomial function.
Given function is known as constant polynomial function.
A function of the form f(x) = c, where c is a real number with a degree of zero at its highest, is known as a constant polynomial. Simply described, a constant polynomial is an algebraic polynomial whose output value does not change regardless of changes made to the input values.
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