Escreva de duas formas diferentes os números a seguir. Urgente
A) 40.000.000.00
B) 0,02
C) 105.000.000
D) 0,000000007
E) 456.983
F) 0,000000673
Answers
Answer:
To solve this problem, we need to write each number in two different ways, we could use percetanges or fractions.
(A) 40,000,000.00.This number can be written in scientific notation like \(4 \times 10^{7}\), where the exponent depends on the amount of zeros the quantity has.
A second way to write this number would be as a product \(2 \times 20,000,000.00\).
(B) 0.02.Similarly, this number can be written in scientific notation \(2 \times 10^{-2}\), also, we can write it as a fraction \(\frac{2}{100}\).
(c) 105,000,000.This number in scientific notation would be \(1.05 \times 10^{8}\). As a product would be
\(1050 \times 100000\).
(d) 0.000000007.In scientific notation would be \(7 \times 10^{-9}\). As a fraction would be \(\frac{7}{1000000000}\).
(e) 456,983.In scientific notation would be \(4.56983 \times 10^{5}\). Also, we can write it as a sum
\(450,000 + 6,983\)
(f) 0.000000673.In scientific notation would be \(6.73 \times 10^{-7}\). As a fraction would be \(\frac{673}{1000000000}\)
Related Questions
The Hiking Club plans to go camping in a state park where the probability of rain on any given day is 0. 66. What is the probability that it will rain on exactly one of the seven days they are there? Round your answer to the nearest thousandth
Answers
The probability that it will rain on exactly one of the seven days the Hiking Club is camping in the state park is approximately 0.293, rounded to the nearest thousandth.
The probability of rain on any given day is 0.66.
To find the probability that it will rain on exactly one of the seven days the Hiking Club is there, we can use the binomial probability formula.
The binomial probability formula is
\(P(x) = C(n, x) * p^x * (1-p)^{(n-x)}\),
where:
P(x) is the probability of exactly x successes,
C(n, x) is the combination formula, which calculates the number of ways to choose x successes from n trials,
p is the probability of success on a single trial, and
n is the total number of trials.
In this case, we want to find the probability of rain on exactly one day out of the seven days.
So, x = 1,
n = 7, and
p = 0.66.
Using the combination formula,
C(n, x) = n! / (x! * (n-x)!),
we can calculate
C(7, 1) = 7! / (1! * (7-1)!)
C(7, 1) = 7.
Plugging the values into the binomial probability formula, we get:
\(P(1) = C(7, 1) * 0.66^1 * (1-0.66)^{(7-1)}\)
\(= 7 * 0.66^1 * 0.34^6\)
Calculating this expression, we find that P(1) is approximately 0.293.
Therefore, the probability that it will rain on exactly one of the seven days the Hiking Club is camping in the state park is approximately 0.293, rounded to the nearest thousandth.
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The following table shows the working populations, average salaries, and income tax rates of France and Germany. Country Workforce Average Salary (€) Income Tax Rate France 31,491,232 29,766 69. 3% Germany 38,844,017 32,104 45. 2% Based on the information in the table, how do the annual tax revenues of Germany and France compare to one another? a. The French government gathers €85,930,190,677 more than the German government. B. The French government gathers €543,141,128,984 more than the German government. C. The German government gathers €309,680,310,056 more than the French government. D. The German government gathers €223,750,119,378 more than the French government.
Answers
Percentage is a way to describe a part of a whole. French government gathers €85,930,190,677 more than the German government.
What is Percentage?A percentage is a way to describe a part of a whole. such as the fraction 1/4 can be described as 0.25 which is equal to 25%.
To convert a fraction to a percentage, convert the fraction to decimal form and then multiply by 100 with the '%' symbol.
We know that the number of people who are working in France is 31,491,232 and their average salary is €29,766 while the amount of tax they pay from their salary is 69.3%. therefore, the total amount of tax that France will gather,
\(Tax_{France} = 31,491,232 \times 29,766 \times 69. 3\%\)
\(= 31,491,232 \times 29,766 \times \dfrac{69. 3}{100}\\\\= 31,491,232 \times 29,766 \times 0.693\\\\=6.49596\times 10^{11}\)
Now, the population that is working in Germany is 38,844,017 while the average earning of the population is €32,104 and they pay 45.2% of their salary as tax. therefore,
\(Tax_{Germany} = 38,844,017 \times 32,104 \times 45.2\%\)
\(= 38,844,017 \times 32,104 \times 45.2\%\\\\= 38,844,017 \times 32,104 \times \dfrac{45.2}{100}\\\\= 38,844,017 \times 32,104 \times 0.452\\\\=5.63666 \times 10^{11}\)
Further, the difference between the tax of the two countries can be written as,
\(\text{Differene in the tax}= Tax_{France} - Tax_{Germany} = 85,930,190,677\)
hence, the French government gathers €85,930,190,677 more than the German government.
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Answer:
Based on the information in the table, how do the annual tax revenues of Germany and France compare to one another?
a. } The French government gathers €85,930,190,677 more than the German government.b. } The French government gathers €543,141,128,984 more than the German government.
c. } The German government gathers €309,680,310,056 more than the French government.
d.} The German government gathers €223,750,119,378 more than the French government.
Step-by-step explanation:
Correct on Edge 2022
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A company that makes square tiles would like to make a square tile that has 4 times the area of their smaller tiles. If their smaller tiles have a perimeter of 16 inches, what will be the perimeter of their larger tile?
Answers
Answer:
The perimeter will be 32 inches
Step-by-step explanation:
- 16 divided by 4 (for each side of square) = 4 inches
- To work out area, we times two sides together 4 * 4
- The area of the smaller tiles is 16 inches squared.
- Times area by 4 for tile that is 4x bigger, 16 * 4 = 64 inches squared. (which you can see is 4 times bigger as 4 * 16 = 64)
- The formula for an area of a square is side * side
- So we need a number that equals 64 when multiplied by itself
- 8 * 8 = 64
- To find perimeter we add up the sides 8 + 8 + 8 + 8 = 32 inches
Maddox is stacking cans to make a display at the grocery store. The bottom row of the stack has 20 cans. Each row has 4 fewer cans than the row below it until there are no cans left.
How many rows of cans are in Maddox’s stack
Answers
Answer: 5
Step-by-step explanation:
Pedro Fernández es gerente de Avon Chic, Pedro estaba interesado en las tasas de rendimiento de los últimos cinco años de dos diferentes fondos mutuos. Trujix, Inc., mostro un periodo de cinco años, tasas de rendimiento del 12; 10; 3; 9 y 11%, mientras que Fondo Corporation arrojo 13; 12:14:10 y 6%. Un cliente se acercó a Pedro y expreso su interés uno de estos fondos mutuos. ¿cuál debería escoger Pedro para su cliente?
Answers
Pedro Fernández should advice his client to go with the second mutual fund named as Fund Corporation. This is because, it showed an higher average return over the five-year period at an average of 11% beating Trujix Inc by 3%.
What is the calculation that supports the above?To get the better fund, Pedro must calculate a simple average of the interest rates of the various funds over the past 5 year period.
Trujix Inc 5-Year Average
(12 + 10+ 3 +9 + 11)/5
= 45/5
=9%
Fund Corporation 5-Year Average
(13 + 12 +14 + 10+ 6)/5
= 55/5
= 11%
Hence, the better investment is the one by Fund corporation.
Full Question
Pedro Fernández is a manager at Avon Chic, Pedro was interested in the rates of return for the last five years of two different mutual funds. Trujix, Inc., showed a five-year period, rates of return of 12; 10; 3; 9 and 11%, while Fund Corporation showed 13; 12:14:10 and 6%. A client approached Pedro and expressed his interest in one of these mutual funds. Which one should Pedro choose for his client?
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Simplify.
75
48
Be sure to write your answer in simplest form.
Answers
Answer:
25/16 or 1 9/16
"The Martin family's truck gets an average of 25 miles per gallon of fuel. Predict how many miles they will be able to get from 7 gallons of fuel by using equivalent rates." (Explain with fractions/ equivalent rate in detail pls)
Answers
Answer:
175 miles
Step-by-step explanation:
Average of 25 miles per gallon of fuel
Miles : gallons = 25 : 1
Predict how many miles they will be able to get from 7 gallons of fuel by using equivalent rates
Let x = number miles
Miles : gallons = x : 7
To find x
25 : 1 = x : 7
25/1 = x/7
Cross product
25*7 = 1*x
175 = x
x = 175 miles
Number miles = 175
A school is having a canned food drive. Each class is challenged to collect 220 cans. If there are x classes in the school, which equation could be used to find the total number of cans, y, the school will collect if each class meets the challenge?
A. y = 20x
B. y = 220x
C. y = 220/x
D. y = 220 + x
Answers
Answer:
B)
Step-by-step explanation:
no. of Classes = x
cans = 220
220×x = 220x
no.of classes × cans = total number of cans
an equilateral triangle and a regular hexagon have perimeters of the same length. if the area of the triangle is 2 square units, what is the area of the hexagon?
Answers
The area of hexagon is found to be 18 square units by using the giving conditions.
What is the equation for calculating a hexagon's perimeter?P = 6a, where an is the length of one of its sides, is the formula for a regular hexagon's perimeter. Here, the hexagon's interior angles are all 120 degrees.
How do you calculate a hexagon's area?The formula for calculating the area of a hexagon is derived from the formula for calculating the area of an equilateral triangle since a regular hexagon is made up of six equilateral triangles. The formula for calculating a hexagon's area is 6(\(\sqrt{3}\)/4)a² where a is the regular hexagon's side length.Given: The perimeters of an equilateral triangle and a regular hexagon are equal.
We need to find the area of the hexagon.
Let,
x = side of trainagle
a = side of hexagon
3x = 6a
x = 2a
Area of equilateral triangle = (\(\sqrt{3}\)/4)x² = 12 square units
Area of hexagon is given as follows:
= 6(\(\sqrt{3}\)/4)a²
= (6\(\sqrt{3}\)/4)(x²/4)
= 12(6)/4
= 18 square units
Therefore, the area of hexagon is found to be 18 square units by using the giving conditions.
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Read each problem carefully. Solve each quadratic equation for the variable(s) specified. Be sure to show all of your work. Explain in two to three sentences what the meof the solution(s) are in relation to the problem situation.1. An object is propelled off of a platform that is 75 feet high at a speed of 45 feet per second (ft./s). The height of the object off the ground is given by the formulah(t) = - 16t^2 + 45t + 75, where h(t) is the object's height at time (t) seconds after the object is propelled. The downward negative pull on the object is represented by -16t^2? Solvefor t.
Answers
Solve for t using quadratic formula:
\(\begin{gathered} t=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \end{gathered}\)Where:
\(\begin{gathered} a=-16 \\ b=45 \\ c=75 \end{gathered}\)so:
\(\begin{gathered} t=\frac{-45\pm\sqrt[]{45^2-4(-16)(75)}}{2(-16)} \\ t=\frac{-45\pm5\sqrt[]{273}}{-32} \\ so\colon \\ t\approx-1.175s \\ or \\ t\approx3.988s \end{gathered}\)We take the positive time, therefore:
Answer:
t = 3.988s
The object will hit the ground after approximately 3.988 seconds.
We can also say that the object will remain in the air for less than 4 seconds.
In 1978, Dawn earned $48000.
A. What was her monthly gross pay?
B. In what month did Dawn reach maximum taxable Social Security income?
C. How much Social Security tax did Dawn pay in February?
D. How much Social Security tax did Dawn pay in May?
E. How much Social Security tax did Dawn pay in November?
Answers
Dawn monthly gross pay is $4000
Dawn reached the maximum taxable Social Security income after 4.425 months
The amount paid on social security tax in February is $242
The amount paid on social security tax in May is $102.85
The amount paid on social security tax in November is zero
What is monthly gross pay?
Monthly gross pay is the amount received by an employee monthly prior to the time at which tax and other deductions are being taken from his/her account.
From the parameters given:
A.
Dawn earned $48000 yearly; her monthly gross pay can be computed as:
\(\\ \\ \\ \mathbf{Monthly \ Gross \ pay = \dfrac{\$48000}{12}}\) (since there are 12 months in a year)
Monthly gross pay = $4000
B.
To determine the month at which Dawn reach the maximum taxable social security income, we need to know the maximum taxable social security income in the year 1978.
From the graph, the taxable income in the year 1978 is $17700.
Thus, the month that Dawn reaches maximum taxable social security income is computed as:
\(\mathbf{=\dfrac{tax. inc \ in \ 1978}{amount \ earned \ per \ month }}\)
\(\mathbf{=\dfrac{\$17700}{\$4000/ month }}\)
= 4.425 months
C.
Looking at the graph in your textbook; in the year 1978, the social security part that was paid in one month is 6.05%.
Therefore, the amount of social security tax paid in Februrary(since it is a single month) is:
\(\mathbf{=\dfrac{6.05}{100}\times \$4000}\)
= $242
D.
In the month of May, the difference between February month and the month of May is 4.
So, if we subtract that from the amount it takes Dawn to reach its max. taxable Social security income, we have:
= 4.425 - 4
= 0.425
Therefore, the amount of social security tax that Dawn pay in the month of May is computed as:
\(\mathbf{=\dfrac{6.05}{100}\times \$4000 \times 0.425}\)
= $102.85
E.
Since the number of months required to complete the maximum social security tax is 4.425 months and the social security tax we are to determine is more than 4.425 months(November); we can conclude that the social security tax Dawn paid for the month of November is zero.
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TEA TIME!!! During a catered lunch, an average of 4 cups of tea are poured per minute. The lunch will last 2 hours. How many gallons of tea should the caterer bring if there are 16 cups in one gallon?
Answers
Answer: 30 gallons
Step-by-step explanation:
First translate 2 hours into 120 minutes. Then multiply 120 * 4 = 480. Then divide 480/16 = 30
In a _______ , _______, not all members of a population have an equal probability of being included?
In an _______, _______, all members of the population have an equal probability of being included.
Some associations are stronger than others, what describes the strength of the association?
A) Effect Size B) Bivariate correlations C) Correlational Samples D) None of the Above
Curvilinear association is one in which the correlation coefficient is zero (or close to zero) and the relationship between two variables isn't a straight line? True/ False
Answers
In a nonprobability sampling, not all members of a population have an equal probability of being included.
In a probability sampling, all members of the population have an equal probability of being included.
The strength of the association is described by the effect size.
Curvilinear association is one in which the correlation coefficient is zero (or close to zero) and the relationship between two variables isn't a straight line. False.
In nonprobability sampling, the selection of individuals from the population is not based on random sampling principles. This means that not all members of the population have an equal probability of being included in the sample.
In probability sampling, every member of the population has an equal and known chance of being selected for the sample. Random sampling methods, such as simple random sampling, stratified random sampling, and cluster sampling, are commonly used to achieve this. In probability sampling, the sample is representative of the population, and statistical inferences can be made.
The strength of the association between two variables is typically measured by the effect size. Effect size quantifies the magnitude or magnitude of the relationship between variables and provides an indication of the practical or substantive significance of the association.
Curvilinear association refers to a relationship between two variables that cannot be adequately described by a straight line. In such cases, the correlation coefficient between the variables may be zero or close to zero, indicating no linear relationship.
Nonprobability sampling involves selecting individuals without an equal probability of inclusion, while probability sampling ensures that all members of the population have an equal chance of being included. The strength of the association between variables is described by the effect size, and a curvilinear association indicates a non-straight line relationship between variables.
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If KM=52 and NL=33, find LM
If the answer is a decimal, round to the nearest tenth
Answers
We need to find LM. Here, we can use the Pythagorean theorem to solve the given problem.Let’s assume, LM = x.We know that the Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
That is,if AC is the hypotenuse, AB and BC are the other two sides of a right triangle ABC, then according to the Pythagorean theorem:AB² + BC² = AC².Now, KM = AB = 52and NL = BC = 33And, LM = AC = x.Using the above information and Pythagorean theorem, we get:AB² + BC² = AC²52² + 33² = x²2704 + 1089 = x²3793 = x²Now, we need to find LM = √x² = x≈61.6 (rounded to the nearest tenth)Therefore, LM ≈ 61.6 (rounded to the nearest tenth).
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The area of a blackboard is 1 1 third square yards. A poster’s area is 8 over 9 square yards. What is the unit rate of the blackboard's area to the poster's area? (plz dont take points)
Answers
Step-by-step explanation:
(1 ⅓ yd²) / (8/9 yd²)
(4/3 yd²) / (8/9 yd²)
(4/3) × (9/8)
3/2
1.5 / 1
Answer:
\(\qquad {:}\longrightarrow\tt 1\dfrac {1}{3}=\dfrac{4}{3}\)
Now answer\(\qquad {:}\longrightarrow\displaystyle\tt \dfrac {\dfrac {4}{3}}{\dfrac {8}{9}}\)
Simplify\(\qquad {:}\longrightarrow\tt \dfrac {4}{3}\times \dfrac {9}{8}\)
\(\qquad {:}\longrightarrow\tt \dfrac {36}{24}\)
\(\qquad {:}\longrightarrow\tt \dfrac {3}{2}\)
\(\qquad {:}\longrightarrow\tt 1.5 \:per\:unit\)
A cable company charges $150 to install equipment. Then there is a $50
charge each month. Write an equation to express this relationship.
Answers
Answer:
Need help with this to
Step-by-step explanation:
A truck delivers 568 oranges to grocery stores each week. How many oranges will it deliver in 25 weeks?.
Answers
568x25=14,200
a manufacturer wishes to set a standard time required by employees to complete a certain process. times from 21 employees have a mean of 6 hours and a standard deviation of 2 hours. test if the mean processing time exceeds 5.5 hours. what is the -value of the test (round off to second decimal place)? assume normal population.
Answers
The -value of the test is 1.90. This indicates that the mean processing time is significantly greater than 5.5 hours, supporting the manufacturer's desire to set a standard time required by employees to complete a certain process
The t-value will be equal to 1.39 for the given statistics.
How to calculate the t-value?To test if the mean processing time exceeds 5.5 hours, we can use a one-sample t-test.
The null hypothesis is that the mean processing time is less than or equal to 5.5 hours:
H0: µ ≤ 5.5
The alternative hypothesis is that the mean processing time is greater than 5.5 hours:
Ha: µ > 5.5
We will use a significance level of 0.05.
The formula for the t-test statistic is:
t = (X - µ) / (s / √n)
Where:
X = sample mean
µ = hypothesized population mean
s = sample standard deviation
n = sample size
Substituting the given values, we get:
t = (6 - 5.5) / (2 / √21) = 1.386
The degree of freedom for the t-test is n-1 = 20.
Using a t-table or calculator, the p-value associated with a t-value of 1.386 and 20 degrees of freedom is 0.093.
Since the p-value (0.093) is greater than the significance level (0.05), we fail to reject the null hypothesis. Therefore, there is not enough evidence to suggest that the mean processing time exceeds 5.5 hours.
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(sin x) (2 sin x cos x)
Answers
Answer:
(sinx) (sin2x)
Step-by-step explanation:
2 sinx cosx = sin 2x
Question in the picture! Answer ASAPP!! Will give brainliest!
Answers
Answer:
A
Step-by-step explanation:
Abdul walks 5 km 35 minutes.how far does he walk in 21 minutes.
Answers
Answer:
Distsnce=speed×time
5×21=105km
Step-by-step explanation:
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Determine the fundamental period of the following signal. Explain your steps in details. x[n]=8+cos(8πn/17)
Answers
Given signalx[n]= 8 + cos(8πn/17)The given signal is a sum of a constant and a cosine signal. The cosine signal is periodic with a period of 17, and a frequency of 8π/17 rad/sample.
To find the fundamental period of the given signal we need to consider both the constant and the cosine signal. Period of constant signal = ∞ Period of cosine signal = 2π/((8/17)π) = 17/4 samples. Now, we need to find the least common multiple (LCM) of the two periods, which will give us the fundamental period.
LCM (17/4, ∞) = 17/4 × 2 = 34/4 = 8.5The fundamental period of the given signal is 8.5 samples. Now, we need to find the least common multiple (LCM) of the two periods, which will give us the fundamental period.
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-3x – y — X
We are doing combining like terms in math and I’m stuck on this question
Answers
Answer:
-4x -y :)
Step-by-step explanation:
Explanation: you would add -3x to x because you are combining like terms and then just subtract the y like usual. Hope I helped :)
What is the distance between points D and F?
A. 5.4
B. 4.2
C. 3.7
D. 2.6
Answers
Answer:
ans is A.............
The required distance between points D and F is 5.4 units.
Coordinate, is represented as the values on the x-axis and y-axis of the graph
Here,
The coordinate of point D is (2, 0) and the coordinate of point F is (-3, -2).
The distance between points D and F can be evaluated by the distance formula,
\(DF = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\)
Substitute the value in the above equation,
\(DF = \sqrt{(2 + 3)^2 + (0 + 2)^2}\\ DF = \sqrt{25 + 4}\\ DF = \sqrt{29}\\ DF = 5.4\)
Thus, the required distance between points D and F is 5.4 units.
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Leticia spends $18.45 on a shirt. She spends at least $3.00 more than Humberto spends. If h represents the amount Humberto spends, which symbol can be used to complete the inequality below to represent this situation?
18.45 blank 3 + h
>
<
Greater-than-or-equal-to
Less-than-or-equal-to
Answers
Answer:
>
Step-by-step explanation:
28.45 >3+hgydnjsjsjjsjkso
Làm giúp em câu 3+4 trắc nghiệm và bài 2+3 ạ
Answers
Tự luận 2a -48/5
2b x= 1/7
x=4/25
match the following. 1. a length equal to the height in a square 2. any rhombus can be created by rotating this shape around the midpoint of its base 3. found by dividing 360 by the number of sides of a regular polygon 4. height of a parallelogram 5. each angle in a rectangle 6. a special type of trapezoid 7. perpendicular distance from the center of a regular polygon to its side 8. a special type of regular polygon
Altitude
base
apothem
parallelogram
isosceles triangle
right angle central angle
Answers
The following points are matched:
A length equal to the height in a square - baseAny rhombus can be created by rotating this shape around the midpoint of its base - right angle (square)Found by dividing 360 by the number of sides of a regular polygon - central angleHeight of a parallelogram - altitudeEach angle in a rectangle - right angleA special type of trapezoid - parallelogramThe perpendicular distance from the center of a regular polygon to its side - apothemA special type of regular polygon - equilateral triangleAll sides of a square are equal.On rotating a square around the midpoint of its base, a rhombus is created. This happens due to the right angles between the sides of the square.The central angle of a regular polygon is obtained by dividing 360 by the number of sides of the same polygon.The altitude of a parallelogram is defined as its height.By the property of a rectangle, it has only right angles.A trapezoid is a quadrilateral with at least one pair of parallel sides while a parallelogram has both pairs of parallel sides.The apothem is the perpendicular distance from the center of a regular polygon to its side.An equilateral triangle is a special type of regular polygon with three equal sides and three equal angles.Thus, the definitions are matched to their respective terms.
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A cyclist is travelling at an average speed of 15km/HR. How long will it take to travel a distance of 52 1/2km
Answers
Answer:
3.5hour
Step-by-step explanation:
Average speed = Distance/Time
Given
Distance = 52 1/2 km
Speed = 15km/hr
Required
Time
Substitute the given values in the formula
Time = (52 1/2)/15
Time= 105/2 * 1/15
Time = 105/30
Time = 3.5hour
hence it will take 3.5hr to travel a distance of 52 1/2km
Let A be a 4x4 matrix over R with characteristic polynomial
(x^4-1) and minimal polynomial (x^2-1). Then
write down all possible rational canonical forms.
Answers
The possible rational canonical forms for the given matrix A are:-
1.
[ 1 1 0 0 ]
[ 0 1 0 0 ]
[ 0 0 -1 0 ]
[ 0 0 0 -1 ]
2.
[ -1 1 0 0 ]
[ 0 -1 0 0 ]
[ 0 0 1 0 ]
[ 0 0 0 1 ]
Let A be a 4x4 matrix over R with characteristic polynomial (x^4-1) and minimal polynomial (x^2-1). To find all possible rational canonical forms, we need to consider the elementary divisors of the matrix A.
The characteristic polynomial gives us the information about the eigenvalues of the matrix A. In this case, the eigenvalues are the roots of the characteristic polynomial, which are 1, -1, i, and -i. Since the minimal polynomial divides the characteristic polynomial, the eigenvalues of the matrix A must satisfy the minimal polynomial as well.
The minimal polynomial, (x^2-1), implies that the eigenvalues of A must be either 1 or -1. Therefore, the eigenvalues i and -i are not valid eigenvalues for this matrix.
Now, let's consider the possible rational canonical forms based on the eigenvalues.
Case 1: Eigenvalue 1
In this case, the Jordan canonical form will have a 2x2 Jordan block corresponding to the eigenvalue 1.
Case 2: Eigenvalue -1
Similar to case 1, the Jordan canonical form will have a 2x2 Jordan block corresponding to the eigenvalue -1.
Hence, the possible rational canonical forms for the given matrix A are:
1.
[ 1 1 0 0 ]
[ 0 1 0 0 ]
[ 0 0 -1 0 ]
[ 0 0 0 -1 ]
2.
[ -1 1 0 0 ]
[ 0 -1 0 0 ]
[ 0 0 1 0 ]
[ 0 0 0 1 ]
These two forms correspond to the two possible ways of organizing the Jordan blocks for the given eigenvalues.
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how to expand and simplify?
\(2( \sqrt{15} - 3 \sqrt{5})^{2} \)
Answers
By the distributive property,
2 (√15 - 3√5)² = 2 (√15 - 3√5) (√15 - 3√5)
… = 2 ((√15)² + 2 (√15) (-3√5) + (-3√5)²)
In other words, we use the identity
(a + b)² = a² + 2ab + b²
with a = √15 and b = -3√5.
Then
2 (√15 - 3√5)² = 2 (15 - 6√(15•5) + 9•5)
… = 2 (15 - 6√(3•5²) + 9•5)
… = 2 (15 - 30√3 + 45)
… = 2 (60 - 30√3)
… = 120 - 60√3