4. Rational, irrational (4 points) (1) (2 points) Prove or disprove that if x y is an irrational number, then x or y is also an irrational number. (2) (2 points) Prove that if x 2 is irrational, then x is irrational. (Hint: try a proof by contrapositive)
Answers
Answer:
See explanation below
Step-by-step explanation:
1) Prove or disprove that if \( x^y\) is an irrational number, then x or y is also an irrational number.
Let's take the following instances:
i) When x= 2 and y=\( \sqrt{2} \) we have: \( 2^\sqrt^{^2^} \)
ii) When \( x=2\sqrt{2} \) and y=3, we have: \( (x=2\sqrt{2})^3 \)
iii) When \( x=2\sqrt{2} \) and \( y = \sqrt{2}\), we have: \( (2\sqrt{2})^\sqrt^{^2^}\)
It is proven because, in scenario
i) x is rational and y is irrational
ii) x is irrational and y is rational
iii) x and y are irrational
2) Prove tha x² is irrational, then x is irrational.
Use contradiction here.
Thus, x² is irrational and x is rational.
\( x =\frac{b}{a} \) when x is rational, a & b are integers.
Therefore, \( x^2 =\frac{b^2}{a^2} \). This x² is rational.
This contradicts the statement that x² is irrational.
Therefore, if x² is irrational, x is also irrational.
Related Questions
8 tens + 3 ones = ____ tens + 13 ones
Answers
Answer:
7 tens
Step-by-step explanation:
the value of the first half of the statement is 83 and the value of the second one is 13 (without the tens) so that mean it has 70 left over to make it 7 tens.
Determine the height of the can of beans(cylinder) given the volume of the
can is 314 in and a radius :of 5 inches. Round to the nearest whole inch.
a. 4 in
b. 20 in
c. 500 in
d. 5 in
Answers
Answer:
4 in(a)
Step-by-step explanation:
height-?
volume-314
radius-5
pie-22÷7 or 3.14
volume= πrsquared h
314=3.14×5×5×h
314= 78.5h
314÷78.5
= 4 in
QUESTION IN PICTURE
Please explain your answer in steps, thank you.
Answers
We can complete the blanks with the following ratios:
(7.5 mi/1) * (1 mi/ 5280 ft) * (400ft/1 yd) * (3 ft/1 ft) =33 flags
Since we do not need a flag at the starting line, then 32 flags will be required in total.
How to obtain the number of flagsTo solve the problem, we would first convert 400 yds to feet and miles.
To convert to feet, we multiply by 3. This gives us: 400 yd * 3 = 1200 feet.
To convert to miles, we would have 0.227 miles.
Now, we divide the entire race distance by the number of miles divisions.
This gives us:
7.5 mi /0.227 mi
= 33 flags
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x = −10, y = −1/5
“Exercises 11-13, the variables x and y vary inversely. Use the given values to
write an equation relating x and y. Then find y when x = 3.”
pls help
Answers
The equation that relates x and y is x = 2/y and the value of y when x is 3 is 2/3
What is inverse variation?Inverse variation is the relationship between two variables, such that if the value of one variable increases then the value of the other variable decreases.
If x varies inversely as y, then ,
x = k/y
k = xy
when x = -10, and y = -1/5
then K = -10 × -1/5 = 2
therefore the equation that relates between x and y is
x = 2/y
when x = 3, y is calculated as;
3 = 2/y
3y = 2
divide both sides by 3
y = 2/3
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Which cause do you think was the most important of the Great Depression why?
Answers
Answer:
The stock market crash of 1929 touched off a chain of events that plunged the United States into its longest, deepest economic crisis of its history. It is far too simplistic to view the stock market crash as the single cause of the Great Depression. A healthy economy can recover from such a contraction.
Step-by-step explanation:
A factory made 3 jars of creamy peanut butter and 97 jars of chunky peanut butter. What percentage of the jars of peanut butter were creamy?
Answers
Answer:
3%
Step-by-step explanation:
They made 97+3 total jars, 3 out of 100 is 3 percent.
A vat of orange contains the juices from 843 oranges . If a company has 89 vats . How many oranges would they use to fill them.
Answers
Alijah had a taxable income of $8450 and filed his federal income tax return with the Single filing status. Using the table below, find the amount he has to pay in taxes.
Single
Taxable income is over
8.350
33 950
82.750
171.550
372,950
But nat over
8,350
33.950
82,250
171.550
372.550
The tax is
Plus
50.00
835.00
4 675.00
16.750.00
41,754.00
108,216.00
10%
15%
25%
20%
33%
35%
Of the
amount over
50
8.350
33.950
82,250
171.550
372,850
O A. $835.00
• B. $850.00
O c. $2087.50
O D. $1252.50
Answers
Answer:
Step-by-step explanation:
67
Alijah's taxable income puts him in the first tax bracket, over $8,350 but not surpassing $33,950. Hence, his tax due is a base of $835 plus 15% of the income that exceeds $8,350. This results in a total tax payable of $850. Option B is the correct answer.
Alijah's taxable income falls within the first bracket of the tax table given, being over $8,350 but not over $33,950.
This bracket requires him to pay a tax of $835.00 plus 15% of the amount over $8,350.
He has a surplus of $100 over the $8,350 threshold (i.e., $8,450 - $8,350), and 15% of $100 equals $15.
Therefore, the total tax he owes would be the fixed amount of $835.00 plus the $15.00 calculated, which sums up to $850.00.
Thus, looking at the provided options, option B ($850.00) would be the correct answer.
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What is the total weight of the bags that weighed /8 pound each?
Answers
The total weight of Rice that Mark buys is given as follows:
2.5 pounds.
How to obtain the total weight?The total weight of Rice that Mark buys is obtained applying the proportions in the context of the problem.
The weight of each bag is given as follows:
5/8 pounds = 0.625 pounds.
The number of bags is given as follows:
4 bags.
Hence the total weight of Rice that Mark buys is given as follows:
4 x 0.625 = 2.5 pounds.
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The half-life of Radium-226 is 1590 years. If a sample contains 400 mg, how many mg will remain after 1000
years?
Answers
Answer:
258.65 mg will be remain after 1000 years
Step-by-step explanation:
This problem can be solved several ways, I prefer the natural log approach for reasons I won’t go into here:
Activity final = Activity initial e^-kt, where
k = the decay constant, = ln2/half life = 0.693/1590, = 0.000436/year
t = the decay time, in the same time base as k
The problem can be worked in units of activity, mass, or number of atoms.
400 x e^-0.000436 x 1000 = 400 x e^-.436 = 258.65 mg
you can check this by estimating. There is about 2/3 of a half life, so there should be more than 200 mg, which would be the value at one half life.
5. In the morning, Kevin's cat eats 1/2 of a can of cat food. In the afternoon, the cat eats 1/4 of a can of food. In the evening, the cat eats 4/5 of a can of food. Which is the most reasonable estimate for the amount of food the cat eats throughout the day: 1 can, 2 cans, or 3 cans?
In just lazy so thanks people who are helping me
Answers
Answer:
2 cans
Step-by-step explanation:
1/2 + 1/4 + 4/5 = 1.55
2 is the best answer since we know the cat eats more than 1 and a half cans so the closest thing would be 2 cans
The tuition T for one year at a college can be approximated by the equation T=1/8x+9 where x is the year since 2000, and T is tuition in thousands of dollars. Use this equation to estimate the tuition for one year at this college in 2020.
Answers
Given that the tuition T for one year is represented by the equation below
\(T=\frac{1}{8}x+9\)Where x is the year since 2000, and T is in thousands of dollars
Statement of Problem: To find tuition for a year in 2020.
In other to find the tuition, we would first find x
Step 1: Find x
\(\begin{gathered} x=2020-2000 \\ x=20 \end{gathered}\)After getting x to be 20, we would then substitute for x in the equation for the tuition
Step 2: Substitute for x
\(\begin{gathered} T=\frac{1}{8}x+9 \\ T=\frac{1}{8}\times20+9 \\ T=2.5+9 \\ T=11.5 \end{gathered}\)After getting T, let us find the value of T in thousands of dollars
Step 3: Find T in thousands of dollars
\(\begin{gathered} T=11.5\times\text{ \$1000} \\ T=\text{ \$11,500} \end{gathered}\)Hence, the tuition T for one year at this college in 2020 is $11,500
11 12 13 14 15 16 17 18 19 20 Given: AB = BC Prove: B is the midpoint of AC. A line contains point A, B, C. A flow chart has 3 boxes that go from top to bottom. The first box is labeled given and contains A B = B C. The second box is labeled definition of congruent segments and contains Line segment A B is-congruent-to Line segment B C. The third box is labeled definition of a midpoint and contains B is the midpoint of Line segment A C. What is the format of this proof? two-column proof one-paragraph proof flowchart proof two-paragraph proof Mark this and return
Answers
A list of assertions and the justifications for each assertion are included in a two-column geometric proof.
What is The Structure of a Proof?Either two columns or a paragraph can be used to write a geometric proof. A two-column proof stated in sentences is all that a paragraph proof is. But since it's simpler to omit stages when writing a paragraph proof, we'll study the two-column technique.A list of assertions and the justifications for each assertion are included in a two-column geometric proof. The reasons why the claims are true are mentioned in the right column, while the statements themselves are listed in the left column. A row in the two-column proof represents each phase of the proof, or each conclusion reached.Make the illustration of the proof-in-use in a figure. Either the figure will be drawn for you already, or you will need to draw it yourself.List the assertions that are made, and then the assertions that must be proven. At this point, the proof has a start and a finish.Based on what you can infer about the figure from the provided details, mark it accordingly. You actually learn how to make the proof at this stage of the proof process, as well as whether or not you can support the assertion that has been stated.Mark any congruent sides, angles, etc. so you can see for yourself what has to be said in the evidence to persuade the reader that you are correct in your conclusion.Without omitting even the simplest, carefully note down each step. The stated statements are frequently among the initial steps (though not always), and the conclusion you want to prove is the last step.To Learn more About geometric proof refer To:
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What is one over two to the fifth power?
Answers
Answer:0.03125
Step-by-step explanation:
GOO*OOGLE
Step-by-step explanation:
(1/2)^5 = 1/2 * 1/2 * 1/2 * 1/2 * 1/2 = 1/32
An amount of $ 8,000 was invested at 6% interest that is compounded yearly. The equation
\(a = 8000 (1.06) ^{t} \)
represents the value A of the investment at time t years.
Find the worth of the investment when t= 7.
Round your answer to the nearest thousand.
Answers
Answer:
The answer is 12000 roundes to the neatest thousand
Step-by-step explanation:
In your calculator just substitute 7 into t
8000 ×1.06 to the power of 7
How do you add radicals and rational exponents?
Answers
Problem:
How do you add radicals and rational exponents?
Solution.
Let the following monomials:
then, if we multiply these two monomials (with the same base), the exponents add up like this:
this is equivalent to:
A. Range
B. Mode
C.mean
D.median
Please help me for a test i will give you a lot of Brainly points
Answers
Answer:
Step-by-step explanation:
A range
help....................
Answers
The unit of the rate of change is (b) dollars per month
How to determine the unit of the rate of changeFrom the question, we have the following parameters that can be used in our computation:
The table of values
Where we have
y = total amount in dollars
x = number of months
The rate of change is calculated as
Rate = y/x
substitute the known values in the above equation, so, we have the following representation
Rate = total amount in dollars/number of months
So, we have
Unit = dollars per month
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what is the number added to 24 and subtracted by 5 the total is 50. find the number
answer it with a explanation pls
Answers
Answer:
31
Step-by-step explanation:
When you add 31 + 24 = 55
55 - 5 = 50
$100,000 is shared among three friends, Anna, Louise and Lacey in the ratio.7: 10:13 respectively. Calculate the amount each receives.
Answers
Answer:
Step-by-step explanation:
Set up an equation:
7x + 10x + 13x = 100000 and solve for x:
30x = 100000 so
x = 3333.33
Anna gets 7(3333.33) = 23333.31
Louise gets 10(3333.33) = 33333.33
Lacey gets 13(3333.33) = 43333.29
what is 5x(23x2)<41 please i need the answer
Answers
Answer:
x < 41/230
Step-by-step explanation:
5x x 46 < 41
230x < 41
Find the distance the point P(3, 1, 7), is to the plane through the three points
Q(5, 0, 3), R(0, -1, 7), and S(7, 5, 0).
Answers
The distance of the point P to the plane is 42/√629 units. The solution has been obtained by using vectors.
What are vectors?
An item with both magnitude and direction is said to contain a vector. An object's motion from one point to another is described by a vector.
We are given three points Q, R and S. So, we will find equation of the plane defined by the three points.
Vector RQ = < 5, 1, -4 > and RS = < 7, 6, -7> and their cross product
RQ x RS = < 5, 1, -4 > x < 7, 6, -7> = < 17 , 7, 23> = n
This is a vector normal to the plane.
So, the equation of the plane is 17x + 4y + 18z = 139
The distance is calculated using
⇒d = (|A·Mx + B·My + C·Mz + D| )/(√A² + B² + C²)
⇒d = (|-17·3 + (-4)·1 + (-18)·7 + 139|)/((-17)² + (-4)² + (-18)²)
⇒d = 42/√629
Hence, the distance is 42/√629 units.
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Can someone solve all this as soon as possible please please
Answers
Answer:
x=405
28.7
x=450
Step-by-step explanation:
A store is having a sale on jelly beans and almonds. For 3 pounds of jelly beans and 5 pounds of almonds, the total cost is $27. For 9 pounds of jelly beans and
7 pounds of almonds, the total cost is $51. Find the cost for each pound of jelly beans and each pound of almonds.
Cost for each pound of jelly beans:
Cost for each pound of almonds:
Answers
Answer:
Cost for each pound of jelly beans: $2.75
Cost for each pound of almonds: $3.75
Step-by-step explanation:
Let J be the cost of one pound of jelly beans.
Let A be the cost of one pound of almonds.
Using the given information, we can create a system of equations.
Given 3 pounds of jelly beans and 5 pounds of almonds cost $27:
\(\implies 3J + 5A = 27\)
Given 9 pounds of jelly beans and 7 pounds of almonds cost $51:
\(\implies 9J + 7A = 51\)
Therefore, the system of equations is:
\(\begin{cases}3J+5A=27\\9J+7A=51\end{cases}\)
To solve the system of equations, multiply the first equation by 3 to create a third equation:
\(3J \cdot 3+5A \cdot 3=27 \cdot 3\)
\(9J+15A=81\)
Subtract the second equation from the third equation to eliminate the J term.
\(\begin{array}{crcrcl}&9J & + & 15A & = & 81\\\vphantom{\dfrac12}- & (9J & + & 7A & = & 51)\\\cline{2-6}\vphantom{\dfrac12} &&&8A&=&30\end{array}\)
Solve the equation for A by dividing both sides by 8:
\(\dfrac{8A}{8}=\dfrac{30}{8}\)
\(A=3.75\)
Therefore, the cost of one pound of almonds is $3.75.
Now that we know the cost of one pound of almonds, we can substitute this value into one of the original equations to solve for J.
Using the first equation:
\(3J+5(3.75)=27\)
\(3J+18.75=27\)
\(3J+18.75-18/75=27-18.75\)
\(3J=8.25\)
\(\dfrac{3J}{3}=\dfrac{8.25}{3}\)
\(J=2.75\)
Therefore, the cost of one pound of jelly beans is $2.75.
1. Ernie earns $17 per hour tutoring student-athletes at Brooklyn University. a. If Ernie tutored for 12 hours this month, how much money did he earn?
Answers
Answer:
$17 * 12 hrs = 204
Step-by-step explanation:
Lauren spends her allowance $50 a month on make up and CDs. She spends $10 more a month on CDs than she does on make up. How much money does she spend every month on each product
Answers
Answer:
$30 on CDs
$20 on Makeup
hope it helps
Step-by-step explanation:
The money spent on each product is $20 and $30 respectively.
How to solve a linear equation?A linear equation can be solved by equating the LHS and RHS of the equation following some basic rules such as by adding or subtracting the same numbers on both sides and similarly, doing division and multiplication with the same numbers.
Given that,
The money spent by Lauren on make up and CDs is $50.
Suppose the money spent on CDs be x.
Then, the money spent on make up is x + 10.
As per the question the following equation can be formed,
x + x + 10 = 50
=> 2x + 10 = 50
=> 2x = 40
=> x = 20
Thus, money spent on make up is 20 + 10 = 30 dollars.
Hence, the money spent on CDs and make up is $20 and $30 respectively.
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what is the answer
3r =30
Answers
Answer:
r=10
Step-by-step explanation:
Answer:
r = 10Step-by-step explanation:
\(3r = 30 \\ \frac{3r}{3} = \frac{30}{3} \\ r = 10 \\ \)
Required Question #7 7) 100^2 x 100^5 = 100^y y=?
Answers
Given the general rule for exponents:
\(a^n\cdot a^m=a^{n+m}\)then, in this case we have the following:
\(\begin{gathered} 100^2x100^5=100^y \\ \Rightarrow(100^2)\cdot(100^5)=100^{2+5}=100^7 \\ \Rightarrow y=7 \end{gathered}\)therefore, y=7
is y=-5/9x proportional
Answers
Answer:
No
Step-by-step explanation:
I don't think it is proportional
Here it is not proportional.
What is proportional?
Proportional relationships are relationships between two variables where their ratios are equivalent.
Another way to think about them is that, in a proportional relationship, one variable is always a constant value time the other.
Here given that,
\(y=-\frac{5}{9x}\)
Here it is not proportional.
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Can someone help me I really need it
Answers
Answer:
No, the figures are similar but not congruent as although width is same, length of fig A is 3 blocks while length of fig. B is 4 blocks. For two fig. to be congruent, their l and b should be eql.
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