In this case, the cube root is = 0
Why is this so?We can use the rule Sarrus to evaluate the determinant
1 w w²
w w² 1
w² 1 w
Starting with the first column, we can wrte out the terms for the diagonal products and
then sum the terms for the products of the diagonal elements that wrap around
(1 x w² x w) +(w x 1 x w²) + (w² x w x w) - (w ² x w x 1) - (w x w² x w) - (1 x w x w ²)
Simplifying each term:
w³ + w³ + w³ - w³ -w³ - w³
We can see that all the terms cancel out, leaving us with a determinant of 0. So it is clear that
|1 w w²|
|w w² 1 |
|w² 1 w | = 0
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Full Question:
Although part of your question is missing, you might be referring to this full question:
Evaluate the determinant 1 w w2 , w w2 1, w2 1 w. where w is a cube root of unity.
Which equations are true for x = –2 and x = 2? Select two options x2 – 4 = 0 x2 = –4 3x2 + 12 = 0 4x2 = 16 2(x – 2)2 = 0
Step-by-step explanation:
x = -2 or x = 2
x + 2 = 0 or x - 2 = 0
(x + 2) (x - 2) = 0
x² - 4 = 0
#CMIIW2.1/0.7=3 but how do you work it out (I cheated and used a calculator)
Answer: To work out 2.1/0.7 without a calculator, you can use long division as follows:
0.7 | 2.1
1.4 (3 times 0.7 is 2.1)
-----
0.7
0.0 (there is no remainder)
Therefore, 2.1/0.7 = 3, which means that 2.1 is equal to 3 times 0.7.
Answer:
see below
Step-by-step explanation:
2.1
-----
.7
Multiply the top and bottom by 10 to get rid of the decimals.
2.1 *10 = 21
.7 * 10 = 7
21/7 = 3
uppose that you are told that the Taylor series of f(x)=x3ex2
about x=0
is
x3+x5+x72!+x93!+x114!+⋯.
Find each of the following:
ddx(x3ex2)∣∣∣x=0=
d7dx7(x3ex2)∣∣∣x=
a. Using Taylor series d(x³eˣ²)/dx about x = 0 is x⁴.
b. Using Taylor series d⁷(x³eˣ²)/dx⁷ about x = 0 is x¹⁰.
What is a Taylor series expansion?A Taylor series is a polynomial expansion of a function about a given point. It is given by f(x - a) = ∑(x - a)ⁿfⁿ(x - a)/n! where
a = point where f(x) is evaluated fⁿ(a) = nth derivative of f(x) about a and n is a positive integerGiven that the Taylor series of the function f(x) = x³eˣ² about x = 0 is
f(x) = x³ + x⁵ + x⁷/2! + x⁹/3! + x¹¹/4!, (1) we proceed to find the given variables
a. To find d( x³eˣ²)/dx about x = 0, the Taylor series expansion about x = 0 is given by
f(x - a) = ∑(x - a)ⁿfⁿ(a)/n!
f(x - 0) = ∑(x - 0)ⁿf(0)/n!
f(x) = ∑xⁿf(0)/n!
f(x) = x⁰f(x)/0! + xf(x)/1! + x²f(x)/2! + x³f(x)/3! + ....
f(x) = f(x) + xf¹(x) + x²f²(x)/2! + x³f³(x)/3! + ....(2)
Since fⁿ(x) is the nth derivative of f(x), and we desire f¹(x) which is the first derivative of f(x). Comparing equations (1) and (2), we have that
x⁵ = xf¹(x)
f¹(x) = x⁵/x
= x⁴
So, d( x³eˣ²)/dx about x = 0 is x⁴.
b. To find d⁷( x³eˣ²)/dx⁷ about x = 0, the Taylor series expansion about x = 0 is given by
f(x - a) = ∑(x - a)ⁿfⁿ(a)/n!
f(x - 0) = ∑(x - 0)ⁿf(0)/n!
f(x) = ∑xⁿf(0)/n!
f(x) = x⁰f(x)/0! + xf(x)/1! + x²f(x)/2! + x³f(x)/3! + ....
Expanding it up to the 8 th term, we have that
f(x) = f(x) + xf¹(x) + x²f²(x)/2! + x³f³(x)/3! + x⁴f⁴(x)/4! + x⁵f⁵(x)/5! + x⁶f⁶(x)/6! + x⁷f⁷(x)/7!.....(3)
Now expanding equation (1) above to the 8th term by following the pattern, we have that
f(x) = x³ + x⁵ + x⁷/2! + x⁹/3! + x¹¹/4! + x¹³/5! + x¹⁵/6! + x¹⁷/7!.....(4)
Since fⁿ(x) is the nth derivative of f(x), and we desire f⁷(x) which is the seventh derivative of f(x). Comparing equations (3) and (4), we have that
x⁷f⁷(x)/7! = x¹⁷/7!
f⁷(x) = x¹⁷/x⁷
= x¹⁰
So, d⁷( x³eˣ²)/dx⁷ about x = 0 is x¹⁰.
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Rewrite the given formula relating the area of a square to the length of its diagonal to solve for d. The 2nd screenshot is the formula sorry for the poor sc
Answer:
\( A=\frac{d^2}{2} \).
Step-by-step explanation:
Area of Square by Applying Formula for the Diagonal
We can use the relationship between diagonal length d and the side a length of a square to find its area A. So, the formula for area of square using diagonal is \( A=\frac{d^2}{2} \).
3.6 is 5% of I need help with all
Answer: 3.6 is 5% of 72
Step-by-step explanation: If 3.6 is 5% then just multiply 3.6 by 95 to get 68.4 and add to get 72, Yw.
100 POINTS!!!
Question
Answer:
A) Sequence 2: 15,13,11,9,7
A) Ordered pairs (12,15), (16,13), (19,11), (22,9), (25,7)
B) Sequence 1: 1,5,17,53,161
B) Sequence 2: 6,15,33,69,141
B) Ordered pairs (1,6), (5,15), (17,33), (53, 69) (161,141)
Step-by-step explanation:
Helping in the name of Jesus.
Answer:
A) Sequence 2: 15,13,11,9,7
A) Ordered pairs (12,15), (16,13), (19,11), (22,9), (25,7)
B) Sequence 1: 1,5,17,53,161
B) Sequence 2: 6,15,33,69,141
B) Ordered pairs (1,6), (5,15), (17,33), (53, 69) (161,141)
Step-by-step explanation:
Helping in the name of typing.
The scatter plot shows the number of households, in millions, that have cable television over eight consecutive years.
Scatter plot with x axis labeled Time in Years and y axis labeled Number of Households with points at 1 comma 3 and 8 tenths, 2 comma 5 and 8 tenths, 3 comma 6 and 2 tenths, 4 comma 7 and 5 tenths, 5 comma 7 and 2 tenths, 6 comma 8 and 3 tenths, 7 comma 9 and 3 tenths, and 8 comma 8 and 5 tenths.
A) y = −2.78x + 33.6
B) y = −2.78x + 48.5
C) y = 2.78x + 33.6
D)y = 2.78x + 48.5
The equaltion for the line of best fit would be,
y = 0.67x + 4.05
The points on the scatter plot are: (1, 3.8), (2, 5.8), (3, 6.2) , (4, 7.5) , (5, 7.2), (6, 8.3), (7, 9.3), and (8, 8.5 )
First we find the mean of the x-values and mean of the y-values.
The mean of the x-values would be,
[tex]\bar{X}[/tex] = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) / 8
[tex]\bar{X}[/tex] = 4.5
and the mean of the y-values would be,
[tex]\bar{Y}[/tex] = (3.8 + 5.8 + 6.2 + 7.5 + 7.2 + 8.3 + 9.3 + 8.5) / 8
[tex]\bar{Y}[/tex] = 7.075
The sum of squares (SSX) = 42
And the sum of products (SP) = 28.2
Regression Equation would be,
y = mx + c
m = SP/SSX
= 28.2/42
= 0.67143
And the y-intercept would be,
c = 4.05357
So, the equation for the line of best fit would be,
y = 0.67143x + 4.05357
Number of Households = 0.67143 (time) + 4.05357
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Find the complete question below.
The scatter plot shows the number of households, in millions, that have cable television over eight consecutive years.
Scatter plot with x axis labeled Time in Years and y axis labeled Number of Households with points at 1 comma 3 and 8 tenths, 2 comma 5 and 8 tenths, 3 comma 6 and 2 tenths, 4 comma 7 and 5 tenths, 5 comma 7 and 2 tenths, 6 comma 8 and 3 tenths, 7 comma 9 and 3 tenths, and 8 comma 8 and 5 tenths.
Find the shortest path from vertex A to vertex L. Give your answer as a sequence of vertexes, like ABCFIL
The shortest path from vertex A to vertex L is ACFIL. The total distance of the path is 27 units.
To find the shortest path from vertex A to vertex L, we can use Dijkstra's algorithm. We start by marking the distance of each vertex from A as infinite except for A, which is 0. Then, we choose the vertex with the smallest marked distance and update the distances of its neighbors. We repeat this process until we reach L.
In this case, we would start at A and update the distances of B and D to 2 and 19, respectively. We would then choose B and update the distances of C, F, and E to 11, 10, and 18, respectively.
Next, we would choose F and update the distances of I and L to 27 and 30, respectively. Finally, we would choose L and have found the shortest path from A to L: ACFIL with a total distance of 27.
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2. Kyle submits a design for the contest, but his explanation was misplaced. How can Figure A be mapped onto Figure B? Can any other transformation be used to map Figure A onto Figure B?
help please i have 5 min
Note that in order to map A onto B, Kyle would have to dilate the given figure by a scale factor or 3.
What is dilation?A dilation is a function f from a metric space M into itself that fulfills the identity d=rd for all locations x, y in M, where d is the distance between x and y and r is some positive real integer. Such a dilatation is a resemblance of space in Euclidean space.
The original point of figure A which has 4 points are
(0,02)
(-1, 2)
(0, 1)
(1, 2)
Multiply all th e points by 3, and you get,
(0,02) x 3 = (0, -6) =
(-1, 2) x 3 = (-3, 6)
(0, 1) x 3 = (0, 3)
(1, 2) x3 = (3, 6)
Plotting the new values will give us the transformation (dilation) required. See the attached image.
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in the figure shown, the darker square is removed. Divide the remaining figure into two rectangles
The dimensions of each rectangle are x by (x- y) and y by (x - y); option A and D
The area of each rectangle is x² - 2xy + y² and x² - xy
The total area of the remaining figure is 2x² - 4xy + 2y²
This figure represents the difference of two squares in that x² - y² = (x + y)(x -y)
What is the are of each triangle?The area of each rectangle is:
x * (x- y) = x² -xy
and
y * (x - y) = xy - y²
The total area of the remaining figure is:
x² -xy + xy - y² = x² - y²
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Rectangle ABCD and A’B’C’D’ are similar.
a. What is the scale factor from ABCD to
A’B’C’D’?
b. What is the scale factor from A’B’C’D’ to
ABCD?
Where Rectangle ABCD and A’B’C’D’ are similar.
a. the scale factor from ABCD to A’B’C’D’ is 1/2
b. What is the scale factor from A’B’C’D’ to ABCD is 2
What is scale factor?In mathematics, a scale factor is the ratio of matching measurements of an item to a representation of that thing. The copy will be bigger if the scaling factor is a full number. The duplicate will be smaller if the scaling factor is a fraction.
When the scale factor is less than one, you are going from big to small. You are dilating negatively.
When you are going from small to big, you scale factor is reversed. In this case, we had 1/2 in a, in b it became 2/1 which = 2.
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Full Question:
Rectangle ABCD and A’B’C’D’ are similar.
a. What is the scale factor from ABCD to A’B’C’D’?
b. What is the scale factor from A’B’C’D’ to ABCD?
See attached image.
Given v = - 5i + 7j and w = - 1 - j a. Find pro*l_{w}*v b. Decompose v into two vectors v_{1} and v_{2} where v_{1} is parallel to w and v_{2} is orthogonal to w
The value of proᵥᵥv is (-30/37)i + (42/37)j and the decomposed vector v into v₁ and v₂ are 6 + 6j and -5i + j - 6 respectively.
a. To find the projection of w onto v (proᵥᵥw), we can use the formula,
proᵥᵥw = (w · ȳ)ȳ where ȳ is the unit vector in the direction of v.
First, let's find the unit vector in the direction of v,
|v| = √((-5)² + 7²) = √(74)
ȳ = v/|v| = (-5/√(74))i + (7/√(74))j
Next, let's find the dot product of w and ȳ,
w · ȳ = (-1)(-5/√(74)) + (-1)(7/√(74))
w · ȳ = 12/√(74)
Finally, we can find the projection of w onto v,
proᵥᵥw = (w · ȳ)ȳ = (12/√(74))((-5/√(74))i + (7/√(74))j)
(w · ȳ)ȳ = (-60/74)i + (84/74)j
(w · ȳ)ȳ = (-30/37)i + (42/37)j
Therefore, the projection of w onto v is (-30/37)i + (42/37)j.
b. v₁ = ((v · w)/|w|²)w
v₂ = v - v₁ where |w| is the magnitude of w. First, let's find |w|,
|w| = √((-1)² + (-1)²)
|w| = √(2)
Next, let's find v · w,
v · w = (-5)(-1) + (7)(-1)
v · w = -12
Using the formula, we can find v₁,
v₁ = ((v · w)/|w|²)w = (-12/2)(-1 - j)
v₁ = 6 + 6j
Now, finding v₂,
v₂ = v - v₁
v₂ = (-5i + 7j) - (6 + 6j)
v₂ = -5i + (7 - 6)j - 6
v₂ = -5i + j - 6
As a result, the vector v can be divided into two parts: v₁ = 6 + 6j and v₂ = -5i + j - 6, where v₁ is parallel to w and v₂ is orthogonal to w.
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Complete question - Given v = - 5i + 7j and w = - 1 - j.
a. Find proᵥᵥv
b. Decompose v into two vectors v₁ and v₂ where v₁ is parallel to w and v₂ is orthogonal to w.
CNNBC recently reported that the mean annual cost of auto insurance is 1016 dollars. Assume the standard deviation is 209 dollars. You take a simple random sample of 94 auto insurance policies. Assuming the original population forms a bell-shaped distribution, answer the following and round your answers to three decimals.
Find the probability that a single randomly selected value is more than 979 dollars.
P(X > 979) = ???
Find the probability that a sample of size
is randomly selected with a mean that is more than 979 dollars.
P(M > 979) = ???
1. The probability that P(X > 979) = 0.567
2. The probability that P(M > 979) = 0.956
How do you calculate probability?
To calculate the probability that a single randomly selected value is more than 979 dollars.
979-1016
= -37/209
= -0.177
P(X > -0.177) = 0.5675
Therefore P(X > 979) = 0.568
To calculate that a sample of size is randomly selected with a mean that is more than 979 dollars.
209 / √94 = 21.557
-37/ 21.557 = -1.7164
P(Z > -1.714) = 0.9564
therefore P(M > 979) = 0.956
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what is 2x to the power of 2
helpppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
Answer:
2x^2 = (2 x 2) x (2 x 2)
= 4 x 4
= 16
I hope this helps you
Solve For X
Solve For Y
Using the properties of a parallelogram, the values of x and y are:
x = 9; y = 22.
How to Find x and y in the Parallelogram?Since the opposite sides are parallel and equal to each other in the parallelogram above, therefore, Opposite angles are equal (or congruent) while the consecutive angles are supplementary to each other.
Therefore, we have:
6y = 180 - 48
6y = 132
Divide both sides by 6:
6y/6 = 132/6
y = 22
(5x + 3) + 6y = 180
5x + 3 + 6(22) = 180
5x + 135 = 180
5x = 180 - 135
5x = 45
x = 9
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Describe the transformation of f(x) to g(x).
A. f(x) is shifted up 1 unit to g(x).
B. f(x) is shifted down pi/2 units to g(x).
C.f(x)is shifted up pi/2 units to g(x).
D. f(x) is shifted up 2 units to g(x).
The transformation of f(x) to g(x) is (a) f(x) is shifted up 1 unit to g(x).
Describing the transformation of f(x) to g(x).From the question, we have the following parameters that can be used in our computation:
The functions f(x) and g(x)
In the graph, we can see that
The graph of g(x) passes through y = 1The graph of f(x) passes through y = 0So, we have
Difference = 1 - 0
Evaluate
Difference = 1
This means that the transformation of f(x) to g(x) is (a) f(x) is shifted up 1 unit to g(x).
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Can someone help me with this pls
Answer:
Subtract the powers, 8-4.
When you simplify that equation itself, the answer is 10,000.
So when you do 10⁴, it's 10,000.
How many more sunflowers with a height of 27 1/2 inches or more were there than sunflowers with a heigh less than 27 1/2 inches?
The number of sun flowers with a height of 27¹/₂ inches or more are 4 more than those with a height of less than 27¹/₂ inches
How to Interpret Dot Plots?A dot plot, which is also known as a strip plot or dot chart, is a simple form of data visualization that comprises of data points plotted as dots on a graph with an x- and y-axis. These types of charts are used to graphically depict certain data trends or groupings.
The number of heights below 27¹/₂ inches that exists in the given dot plot is seen to be 8 in number.
Similarly, the number of dot plots that exists above or equel to 27¹/₂ inches from the given dot plot are 12 in number.
Therefore, we can say that:
Difference in total number for both parameters = 12 - 8= 4
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Determine whether y=9(-5)^x represents an exponential function
Answer: Yes, y = 9(-5)^x represents an exponential function.
An exponential function is a function in the form y = ab^x, where a and b are constants and b is a positive real number not equal to 1. The base, b, is raised to the power of x, and the resulting value is multiplied by the constant a.
In the given function, we have a = 9 and b = -5. Although b is negative, it is still a real number not equal to 1, so the function is still considered exponential. Furthermore, the exponent x is a variable, which is raised to a constant base -5, meeting the definition of an exponential function.
Therefore, y = 9(-5)^x is an exponential function.
Step-by-step explanation:
which fraction is the smallest? 3to4, 2/3, 10:12, 2to1
The smallest fraction is 8/12, which is equivalent to 2/3.
How to determine the smallest fractionTo compare fractions, it's best to convert them all to a common denominator.
3/4 is equivalent to 9/12
2/3 is equivalent to 8/12
10/12 is equivalent to 5/6
2/1 is equivalent to 24/12
So, the smallest fraction is 8/12, which is equivalent to 2/3.
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A meter in a taxi calculates the fare using the function f(x) = 2.56x + 2.40. If x represents the length of the trip, in miles, how many miles can a passenger travel for $20?
A passenger can travel 6.875 miles for $20 using the given taxi fare function.
To determine how many miles a passenger can travel for $20 using the taxi fare function f(x) = 2.56x + 2.40, we need to set up an equation and solve for x.
The equation we need to solve is:
2.56x + 2.40 = 20
To solve for x, we can start by subtracting 2.40 from both sides of the equation:
2.56x = 17.60
Next, we can divide both sides by 2.56:
x = 6.875
This calculation assumes that the fare is based solely on distance and that there are no additional fees or charges.
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pls help quickly!!
Factor completely
3zy²x + y²x - 12zx - 4x
Select one:
a. x (3z + 1) (y + 2)(y-2)
b. None of these.
c.xy (3z + 1)(y-2)
d. x (3z + 1) (4z-3)
e. x (3z-1) (y + 2)²
The volume of a cube depends on the length of its sides. This can be written
in function notation as v(s). What is the best interpretation of v(4) = 64?
A. 4 sides of the cube have a total length of 64 feet.
B. A cube with side lengths of 4 feet has a volume of 64 cubic feet.
C. 4 of these cubes will have a total volume of 64 cubic feet.
OD. A cube with a volume of 4 cubic feet has side lengths of 64 feet.
Answer:
C is the correct answer.
(-2-3)-(4-5)+ (−1+6)−(−9+8)=
Answer:
Step-by-step explanation:
(-2-3)-(4-5)+ (−1+6)−(−9+8)
-5-4+5-1+6+9-8
-9+4+15-8
-5+7
2
what is five term in 12,16,21,27
Answer:
34
Step-by-step explanation:
This is sequnce has an easy pattern with adding 1 to the difference every time and goes +4, +5, +6 , so the next should be +7
Answer: think it’s a5=34
Step-by-step explanation:
Select all that is TRUE
Question 11 options:
Adding a constant to all the data points does NOT change the correlation
Interchanging X and Y changes the correlation
Correlation has no units
Outliers do NOT effect the correlation
If r is close to 1 then it is a strong positive correlation
Correlation could be positive or negative
Correlation and y-intercept have the same sign
The True statements about the correlation are: (1), (2), (3), (5), (6) and (7)
1) We know that the adding, subtracting, multiplying or dividing a constant to all of the data points in one or both variables does not change the correlation coefficient.
So, the first statement is True.
2) We know that the correlation measures the strength of the pattern around a line of regression as well as the direction of the line. So, interchanging the variables would not change the correlation. The correlation between x and y is the same as the correlation between y and x.
Thus the second statement is True.
3) We know that the correlation coefficient does not have any units.
It is just a number between -1 to +1.
Thus the third statement is True.
4) We know that the outlier may weaken the correlation. It makes the data more scattered. So the correlation coefficient r gets closer to 0.
Thus the fourth statement is False.
5) We know that if the correlation coefficient r is close to 1 then it is a strong positive correlation and if r is close to -1 then it is a strong negative correlation.
Thus the fifth statement is True.
6) We know that the corrletaion coefficient values can range from -1 to 1.
Thus the sixth statement is True.
7) We know that the y-intercept is nothing but the point at which the regression line intercepts the y-axis. The y-intercept of the regression line and the correlation coefficient have the same sign.
Thus the seventh statement is True.
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Passersby have taken 4 pieces of vanilla cake and 6 pieces of coconut cake from a sample tray. Based on past data, of the next 30 samples taken, how many should you expect to be pieces of vanilla cake?
Answer: The answer is actually 12 pieces of vanilla cake.
The combinations for the lockers at school consist of 3 numbers. Each number in the combination can be a number from 0-29. How many locker combinations are possible?
Answer:
87 I think.
Step-by-step explanation:
29 x 3 = 87
In the diagram above, the sine of angle ACB = 3/5. What is the sine of angle ANM?
The sine of the angle ANM is equal to 3/5 as the triangles are similar.
How to calculate for the sine of angle ANM in the triangle.Given that the sine of angle ACB is equal to 3/5, line MN is parallel to BC and MN is midsegment of triangle ABC which implies that the triangles ABC and ANM are similar so;
sine ANM = 3/2 ÷ 5/2
sine ANM = 3/2 × 2/5
sine ANM = 3/5.
Therefore, the sine of the angle ANM is equal to 3/5 as the triangles are similar.
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29 Given: A = √363 and B = √27
Explain why A + B is irrational.
Explain why A B is rational.
To explain why A + B is irrational, we need to show that it cannot be expressed as a ratio of two integers.
Suppose that A + B is rational, which means we can write it as the ratio of two integers p and q (where q is not zero):
A + B = p/q
Now, we can substitute the values of A and B and simplify:
√363 + √27 = p/q
We can then rearrange the terms to isolate one of the square roots:
√363 = p/q - √27
We can square both sides of this equation to eliminate the square roots:
363 = p^2/q^2 + 27 - 2(p/q)√27
Notice that the right-hand side of this equation has a term with a square root. This means that if we assume that A + B is rational, we arrive at a contradiction: we have shown that √363 (which is equal to A) is irrational, which means that p^2/q^2 + 27 must also be irrational. However, the left-hand side of the equation is rational. Therefore, our assumption that A + B is rational must be false, and we conclude that A + B is irrational.
To explain why AB is rational, we can use the fact that the product of two rational numbers is rational.
We can rewrite A and B as follows:
A = √(363) = √(121 x 3) = √(11^2 x 3) = 11√3
B = √(27) = √(9 x 3) = √(3^2 x 3) = 3√3
Therefore, AB = 11√3 x 3√3 = 33 x 3 = 99.
Since 99 is a rational number (which can be expressed as the ratio of the integers 99 and 1), we conclude that AB is rational.