Show that if a, b, and m are integers such that m ≥ 2 and a ≡ b (mod m), then gcd(a, m) = gcd(b, m).
Answers
This means that gcd(a, m) is a subset of gcd(b, m) (since any common divisor of a and m is also a common divisor of b and m), and similarly, gcd(b, m) is a subset of gcd(a, m). Therefore, gcd(a, m) = gcd(b, m).
To show that gcd(a, m) = gcd(b, m) when a ≡ b (mod m) and m ≥ 2, we can use the fact that if d divides both a and m, then it also divides b (since a ≡ b (mod m) implies that m divides a-b).
So, let's start by letting d be a common divisor of a and m, and let's show that it is also a common divisor of b and m. Since d divides a and m, we can write a = kd and m = ld for some integers k and l. Then, we have:
b ≡ a (mod m) (by the definition of congruence)
b ≡ kd (mod ld) (substituting a = kd and m = ld)
b = jd (where j = k mod l, since ld divides kd and hence j is an integer)
Therefore, we have shown that if d divides both a and m, then it also divides b and m.
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Related Questions
what is greatest common denominator of 72 and 159)
Answers
Answer:
3
Step-by-step explanation:
The greatest common divisor (GCD) of two nonzero integers a and b is the greatest positive integer d such that d is a divisor of both a and b (Wikipedia)
To find out which is the GCD of 72 and 159, keep factoring both until they cannot be factored any more
Then take the factors common to both list of factors; the greatest among them is the GCD
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
The factors of 159 are: 1, 3, 53, 159
The common factors are 1, 3
So GCD is 3
Which of the following would NOT be a valid way to summarize or visualize a categorical variable? Choose the correct answer below
Bar chart Pie chart Relative frequency table None of the above
Answers
All of the methods mentioned, i.e., bar chart, pie chart, and relative frequency table are valid ways to summarize or visualize categorical variables. Option D.
They are commonly used in data analysis to gain insights into the distribution and proportion of different categories within a dataset.
A bar chart is a graphical representation of data that uses rectangular bars to display the frequency or proportion of different categories. It is useful in comparing the frequencies of different categories and identifying the most common or rare categories.
A pie chart is another graphical representation of data that uses slices of a circle to display the relative frequency or proportion of different categories. It is useful in showing the proportion of each category in relation to the whole.
A relative frequency table is a tabular representation of data that displays the frequency and proportion of each category. It is useful in comparing the frequencies and proportions of different categories and identifying the most common or rare categories.
Therefore, none of the options given would be an invalid way to summarize or visualize categorical variables. The choice of which method to use depends on the nature of the data and the purpose of the analysis.
It is important to choose a method that effectively communicates the information being presented and is appropriate for the audience. So Option D is correct.
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Note the correct question is
Which of the following would NOT be a valid way to summarize or visualize a categorical variable? Choose the correct answer below
A.)Bar chart
B.)Pie chart
C.)Relative frequency table
D.)None of the above
Let f(x)=x√x-4
What is the average rate of change of f(x) from 8 to 14?
Answers
Answer:
Avg rate of change =
[ f(14 ) - f(8) ] / [ 14 - 8 ] =
[ ( √ [14 - 4] ) - (√ [8 - 4]) ] / 6 =
[ √10 - √4 ] / 6 =
0.19
It can sometimes get colder than 3 degrees in the north. Select all the
possible temperatures for the north.
A) -2 degrees
B) 0 degrees
C) 2 degrees
D) 4 degrees
E) 6 degrees
Answers
The possible temperatures for the north will be 4 degrees and 6 degrees. The correct option is D and E.
How to calculate the temperature?Mathematical statements with at least two terms connected by an operator and containing either numbers, variables, or both are referred to as expressions.
The direction that heat energy will spontaneously flow depends on temperature. Temperature is the degree to which a body is hot or cold. because it was mentioned that the north can occasionally experience temperatures below 3 degrees. D and E are the appropriate choices
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a sixth grade class of 20 students must raise $298 to go on a field trip. what equation can be used to find m, the amount of money that each student must raise?A.298 - 20 = mB.298 m = 20C.298 (20) = mD.20 m =298
Answers
Let's begin by listing out the given information:
Number of students (n) = 20
Amount (a) = $298
The equation that can be used to find the amount of money that each student must raise (m) is given below:
\(\begin{gathered} m=\frac{298}{20} \\ \Rightarrow20\cdot m=2988 \\ \therefore20m=2988 \end{gathered}\)Hence, option D is the correct answer
The rectangular floor of a classroom is 20 feet in length and 30 feet in width.
A scale drawing of the floor has a length of 4 inches. What is the area, in
square inches, of the floor in the scale drawing?
Answers
Answer:
Area = 600
Step-by-step explanation:
The area formula for rectangles is length x width. In this equation, the length is 20 and the width is 30, multiplying them together gets us the area of 600.
given in this theorem to evaluate the integral. 2 9x dx Find the width of each subinterval in terms of n. units Find the ith endpoint in terms of n. X;= Evaluate f(x) = 9x at the ith endpoint. f(x;) = Evaluate the integral.
Answers
To evaluate the integral ∫2^9x dx, we can use the definite integral formula. Let's divide the interval [2, 9] into n subintervals of equal width.
The width of each subinterval can be found by taking the difference between the endpoints of the interval and dividing it by the number of subintervals: Δx = (9 - 2)/n. The ith endpoint, denoted as xi, can be determined by multiplying the width of each subinterval by i and adding it to the lower endpoint: xi = 2 + iΔx. We can evaluate f(xi) = 9xi at the ith endpoint by substituting xi into the function. Finally, we can evaluate the integral using the definite integral formula: ∫2^9x dx = lim(n→∞) Σ[i=1 to n] f(xi)Δx.
In summary, to evaluate the integral ∫2^9x dx, we divide the interval [2, 9] into n subintervals of equal width. The width of each subinterval is given by Δx = (9 - 2)/n. The ith endpoint, xi, is determined by multiplying the width of each subinterval by i and adding it to the lower endpoint. We evaluate f(xi) = 9xi at the ith endpoint by substituting xi into the function. Finally, we can find the integral by taking the limit of the sum of f(xi)Δx as n approaches infinity.
Now let's explain the steps in more detail. We start by finding the width of each subinterval. The interval [2, 9] has a difference of 7, and we divide it into n equal subintervals. Thus, the width of each subinterval is Δx = (9 - 2)/n.
Next, we determine the ith endpoint, xi, for each subinterval. We multiply the width of each subinterval, Δx, by i and add it to the lower endpoint of the interval. In this case, the lower endpoint is 2. Therefore, xi = 2 + iΔx.
To evaluate f(xi) = 9xi at the ith endpoint, we substitute xi into the function f(x) = 9x. This gives us f(xi) = 9(2 + iΔx).
Finally, we can evaluate the integral by taking the limit of the sum of f(xi)Δx as the number of subintervals, n, approaches infinity. The integral is expressed as ∫2^9x dx = lim(n→∞) Σ[i=1 to n] f(xi)Δx.
By following these steps and taking the limit as n approaches infinity, we can evaluate the integral.
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should the researcher use the rows or the columns of the field as blocks? justify your answer.
Answers
Yes, the researcher use the rows or the columns of the field as blocks.
When conducting research, it is important to think about which structure to use as the basis for your work.
In particular, when dealing with a matrix, the question arises of whether the researcher should use the rows or columns as blocks.
Both have their own advantages and drawbacks, which should be taken into consideration before making a decision.
However, it can be challenging to discover similarities between data points within the same row.
Additionally, the matrix structure can become distorted, as the data points no longer form a neat square or rectangle.
Ultimately, which approach the researcher should take depends on the type of research being conducted and the goals of the project.
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In a certain board game, a player rolls two fair six-sided dice until the player rolls doubles (where the value on each die is the same). The probability of rolling doubles with one roll of two fair six-sided dice is.
Answers
There are 36 possible combinations from rolling two dice.
There are 6 combinations of doubles that could be rolled.
6/36 = 1/6 ≈ 0.1667 or about 16.67%.
Carlisle Transport had $4,520 cash at the beginning of the period. During the period, the firm collected $1,654 in receivables, paid $1,961 to supplier, had credit sales of $6,916, and incurred cash expenses of $500. What was the cash balance at the end of the period?
Answers
To calculate the cash balance at the end of the period, we need to consider the cash inflows and outflows.
Starting cash balance: $4,520
Cash inflows: $1,654 (receivables collected)
Cash outflows: $1,961 (payments to suppliers) + $500 (cash expenses)
Total cash inflows: $1,654
Total cash outflows: $1,961 + $500 = $2,461
To calculate the cash balance at the end of the period, we subtract the total cash outflows from the starting cash balance and add the total cash inflows:
Cash balance at the end of the period = Starting cash balance + Total cash inflows - Total cash outflows
Cash balance at the end of the period = $4,520 + $1,654 - $2,461
Cash balance at the end of the period = $4,520 - $807
Cash balance at the end of the period = $3,713
Therefore, the cash balance at the end of the period is $3,713.
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Part A
In the table, describe the shape of the cross section formed when a particular plane passes through the cone..
A
Description of Plane
plane parallel to the circular base, not passing through the tip of the cone
plane parallel to the circular base, passing through the tip of the cone
plane not parallel to the base, not passing through the base, and making an angle with the horizontal that is less than that made by the
slant height of the cone
plane making an angle with the horizontal that is greater than that made by the slant height, passing through the tip of the cone
Description of Cross
Section
Answers
All the solutions are;
1) Circle
2) A point
3) An oval that becomes more elongated as the angle with the horizontal increases.
4) An isosceles triangle.
Since, A cross-section is a plane section that is a section of a three-dimensional object that is parallel to one of its planes of symmetry or perpendicular to one of its lines of symmetry.
Now, We can formulate;
1) Plane Parallel to the circular base, not passing through the tip of the cone:
⇒ Circle.
2) Plane Parallel to the circular base, passing through the tip of the cone:
⇒ A point.
3) Plane not parallel to the base, not passing through the base, and making an angle with the horizontal that is less than that made by the slant height of the cone:
⇒ An oval that becomes more elongated as the angle with the horizontal increases.
4) Plane making an angle with the horizontal that is greater than that made by the slant height, passing through the tip of the cone :
⇒ An isosceles triangle.
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Find the Present Value of Perpetuity that pays you $1,800 per
year forever assuming your money is worth 5%?
* Please be very detailed in your answer.
Answers
Therefore, the present value of the perpetuity that pays $1,800 per year forever, assuming a 5% interest rate, is $36,000. To find the present value of a perpetuity that pays $1,800 per year forever, we can use the formula:
Present Value = Cash Flow / Interest Rate
In this case, the cash flow is $1,800 and the interest rate is 5%. Plugging these values into the formula, we get:
Present Value = $1,800 / 0.05. Simplifying this equation, we find that:
Present Value = $36,000
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Can someone Help me with this question ASAP!
Answers
Answer:
A =5
Step-by-step explanation:
From the Pythagorean theorem
Answer:
The perimeter is 30ft
Step-by-step explanation:
\(a^2+12^2=13^2\\a^2=13^2-12^2\\\sqrt{a^2}=\sqrt{25}\\a=5\\\\5+12+13=30\)
What is the radius of a sphere with a volume of 32398cm3 cm
to the nearest tenth of a centimeter?
Answers
Answer:
The answer is 19.8cm to the nearest tenth
f(x)=log5x what Is the range of the function
Answers
The range of the function f(x) = log5x is (-∞, +∞).The function f(x) = log5x represents the logarithm base 5 of x. To determine the range of this function, we need to consider the possible values that the logarithm can take.
The range of the logarithm function y = log5x consists of all real numbers. The logarithm function is defined for positive real numbers, and as x approaches 0 from the positive side, the logarithm approaches negative infinity. As x increases, the logarithm function approaches positive infinity.
The range of the function is the set of all possible output values. In this case, the range consists of all real numbers that can be obtained by evaluating the logarithm
log5(�)log 5 (x) for �>0 x>0.
Since the base of the logarithm is 5, the function log5x will take on all real values from negative infinity to positive infinity. Therefore, the range of the function f(x) = log5x is (-∞, +∞).
In other words, the function can output any real number, ranging from negative infinity to positive infinity. It does not have any restrictions on the possible values of its output.
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Answer: All real numbers
Step-by-step explanation:
Edge
There were 3 squirrels and 18 acorns on Josh's back porch. What was the ratio of acorns to squirrels? Enter your answer in reduced form.
6:1
18:3
3:18
1:6
Answers
Answer:
1/6
Step-by-step explanation:
determine whether the variable is qualitative or quantitative. model of car driven
Answers
(a) The complex Fourier series of f(t) is given by ∑(n=-∞)^(∞) c_n exp(jnωt), where c_n = { 6j/(7πn) if n is odd, 0 if n is even }.
(b) The trigonometric Fourier series of f(t) is given by ∑(n=0)^(∞) [a_n cos(nωt) + b_n sin(nωt)], where a_n = 0 for all n, and b_n = { 12/(nπ) if n is odd, 0 if n is even }.
(a) To determine the complex Fourier series of f(t), we first need to find the coefficients c_n. The complex Fourier series representation is of the form ∑(n=-∞)^(∞) c_n exp(jnωt), where ω = 2π/T is the fundamental frequency.
For the given function f(t), we have the following recursive relationship:
f(t) = 4t + 6f(t+7)
To find c_n, we need to compute the Fourier coefficients. Multiplying both sides of the recursive relationship by exp(-jmωt) and integrating over one period T, we get:
∫[0]^[T] f(t) exp(-jωnt) dt = ∫[0]^[T] (4t + 6f(t+7)) exp(-jωnt) dt
Expanding the integral on the right-hand side using the linearity property of the integral, we have:
∫[0]^[T] f(t) exp(-jωnt) dt = 4∫[0]^[T] t exp(-jωnt) dt + 6∫[0]^[T] f(t+7) exp(-jωnt) dt
The first integral on the right-hand side can be evaluated using integration by parts. The second integral involves the function f(t+7), which has a periodicity of 7. Thus, we can rewrite it as:
∫[0]^[T] f(t+7) exp(-jωnt) dt = ∫[7]^[T+7] f(t) exp(-jωnt) dt
Substituting these results back into the equation and simplifying, we get:
c_n = 4(∫[0]^[T] t exp(-jωnt) dt) + 6(∫[7]^[T+7] f(t) exp(-jωnt) dt)
Now, we need to evaluate the integrals. The first integral can be computed using integration by parts or by recognizing it as the Fourier coefficient of t. The result is:
∫[0]^[T] t exp(-jωnt) dt = jT/(nω)^2
The second integral can be simplified using the periodicity of f(t+7):
∫[7]^[T+7] f(t) exp(-jωnt) dt = ∫[0]^[T] f(t) exp(-jωn(t+7)) dt = exp(-j7nω) ∫[0]^[T] f(t) exp(-jωnt) dt
Since f(t) has a periodicity of 7, the integral becomes:
∫[7]^[T+7] f(t) exp(-jωnt) dt = exp(-j7nω) ∫[0]^[7] f(t) exp(-jωnt) dt
Substituting these results
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Distance Conversions The United States, Liberia, and Myanmar are the only three countries that have not adopted the metric system as its primary system of measurement. There have been reports and legislation calling for a conversion to the metric system since the 1960s. The U.S. Department of Commerce's National Institute of Standards and Technology wrote that "industrial and commercial productivity, mathematics and science education, and the competitiveness of American products and services in world markets, will be enhanced by completing the change to the metric system of units" (1997, 2). From a student's perspective, it would be better if the U.S. converted to the metric system. Because scientists globally use the metric system, most of U.S. scientists do, as well. Until the U.S. "metrifies", students in the United States will need to continue to use and understand both measurement systems. 6. Refer to Appendix 2.1. Show your calculations for the following questions. 33 | Lab 2: Map Interpretation a. 2 miles contain how many feet? b. 1 mile contains how many inches? c. 10 kilometers contain how many meters? d. 1 kilometer contains how many centimeters? e. How many miles are in a "5k" (five-kilometer) race? f. A marathon is 26.2 miles. How many kilometers is this?
Answers
Answer:
a. 10560
b. 63360 in.
c. 10000 m
d. 100000 cm
e. 3.107 miles
f. 42.165 km
Step-by-step explanation:
a. 2 miles contain how many feet?
1 mile = 5280 ft
2 miles × 5280 ft / mile = 10560 ft
b. 1 mile contains how many inches?
1 ft = 12 in
1 mile × 5280 ft/mile × 12 in./ft = 63360 in.
c. 10 kilometers contain how many meters?
1 km = 1000 m
10 km × 1000 m/km = 10000 m
d. 1 kilometer contains how many centimeters?
1 km × 1000 m/km × 100 cm/m = 100000 cm
e. How many miles are in a "5k" (five-kilometer) race?
1 in. = 2.54 cm
5 km =
= 5 km × 1000 m/km × 100 cm/m × 1 in. / (2.54 cm) × 1 ft / (12 in.) × 1 mile / (5280 ft)
= 3.107 miles
f. A marathon is 26.2 miles. How many kilometers is this?
26.2 miles =
26.2 miles × 5280 ft / mile × 0.3048 m/ft × km / (1000 m) = 42.165 km
point) The number of Drosophila fruit flies in controlled experiment assumes the following growth pattern when the food source is limited: 600 N(t) 1+ Me-0.3t where measured in days. A. How many fruit files were there in the beginning? What is the limiting number of fruit flies 600 C.At what time was the population increasing most rapidly? D. At what rate is the number of fruit flies increasing after days? 600/(1+11e
Answers
time was the population increasing most rapidly. At rate is the number of fruit flies increasing after days A. 600 B. 600 C. t=0 D. 0.09
The equation 600/(1+Me-0.3t) is a model for the population of Drosophila fruit flies, with time representing the number of days the experiment has gone on for. A. The population of fruit flies in the beginning is given by the equation as 600. B. The limiting number of fruit flies is also given by the equation as 600. C. The population is increasing most rapidly when t=0. D. After days, the number of fruit flies is increasing at a rate of 0.09, which can be calculated by taking the derivative of the equation with respect to t.
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PLS HELP WITH BOTH SCREEN SHOTS I WILL MARK BRAINLIEST
Answers
Answer:
Answer 1. 3in
Answer 2. 0.875in
Answer 3. 26 miles
Answer 4. 7.67in
Step-by-step explanation:
Answer 1. We know that 2 feet in real life is going to equal 1 inch, so since the person is 6 feet, we divide that by 2 to get 3. So the action figure is going to be 3in.
Answer 2. In this case, we see that 1 foot is equal to four inches. That must mean that 1 inch is equal to a quarter foot. So we just multiply
0.25 x 3.5 = 0.875
The length is 0.875 feet
Answer 3.
We know that 1 inch = 6.5 miles, and there are 4 inches on the map, so we have to multiply 6.5 x 4 to find out how many miles there is.
Answer 4.
To find this one out, we have to divide 9 into 69. That gives us 7.67. So 69 nautical miles would be equal to 7.67 inches on the map
Hope this helps you out :)
Answer:
ty
Step-by-step explanation:
How do you solve an inequality with 3 sides?
Answers
help me solve the blanks pleaseeee
Answers
Answer: hi
Step-by-step explanation:
EASY 100 POINTS, WILL MARK FIRST ANSWER BRAINLIEST
The system of linear equations −3x + 2y = −6 and y equals negative one half times x plus 4 is graphed on a coordinate plane. Approximate the solution to the system.
(−1.5, −3.5)
(−1.5, 4.25)
(2.5, 1.5)
(3.5, 2.25)
Answers
Answer:
D. (3.5, 2.25)
Step-by-step explanation:
A solution to the system is where the two equations intersect.
looking at the graph, we see that the intersection point is between x= 3 and 4, and Y = 2 and 3.
Answer:
(3.5, 2.25)
Step-by-step explanation:
To find the coordinate of the solution to the system of linear equations −3x + 2y = −6 and y = -1/2x + 4, we need to solve the system. One method to do this is substitution. We can solve one of the equations for y, then substitute the expression for y into the other equation. This will give us an equation in terms of x, which we can then solve for x. Then, we can substitute the value of x back into one of the equations to find the corresponding value of y.
Using substitution, we can solve the system as follows:
y = -1/2x + 4
-3x + 2y = -6
Substitute y = -1/2x + 4 into -3x + 2y = -6:
-3x + 2(-1/2x + 4) = -6
-3x + (-x + 8) = -6
-4x + 8 = -6
-4x = -14
x = 3.5
Substitute x = 3.5 into y = -1/2x + 4:
y = -1/2(3.5) + 4
y = 2.25
Therefore, the solution to the system is approximately (3.5, 2.25).
An 8-pack of board games costs $34.00. What is the unit price?
Answers
Step-by-step explanation:
Each unit would be $4.25
Answer:
8/$34.00 divided by 8 = 1/$4.25. 1 board game costs $4.25. hope this helps
Step-by-step explanation:
- Zombie
What transformation of the parent function f(x) is made to get f(3x)?
Answers
Answer: Vertically shifting it by 3
The transformation of a function is horizontally shrink by a factor of 3 .
What is a horizontal shrink?We can apply horizontal shrink to a function by multiplying its input values by a scale factor, a, where 0 < 1/a < 1.Let’s go ahead and look at how f(x) = x2 will be affected by a scale factor of 1/2 and 1/3.
Below is a graph of the data.As we have expected, the graph stretches by a factor of 2 and 3. This is true for all horizontal stretches. The graph only stretches away from the y-axis when we horizontally stretch a graph.Horizontal stretch on other functions will exhibit similar properties. Let’s say we have f(x) = |x|, if this function’s graph is to be stretched horizontally to attain g(x), the new function’s expression can be expressed as |1/3 ∙ x| = |x/3|.How do you horizontally shrink by a factor of 3 ?
If g(x) = 3f (x): For any given input, the output of g is three times the output of f, so the graph is stretched vertically by a factor of 3. If g(x) = f (3x): For any given output, the input of g is one-third the input of f, so the graph is shrunk horizontally by a factor of 3.Learn more about Transformation on :
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Laboratory tests show that the lives of
light bulbs are normally distributed with
a mean of 750 hours and a standard
deviation of 75 hours. Find the
probability that a randomly selected
light bulb will last between 750 and 825
hours.
[? ]%
Answers
Answer:
34%
Step-by-step explanation:
LammettHash is right just take it as a whole number (for those of you using acellus)
a rectangular tank that is 1372 ft3 with a square base and open top is to be constructed of sheet steel of a given thickness. find the dimensions of the tank with minimum weight
Answers
The dimensions of the tank with minimum weight is height= 7 and width and length= 14.
The minimum surface area implies that the tank has minimum weight.
Given that the base is square, its width is equal to its length.
Let x= the base width and length of the tank.
Let h = the height .
We are given the volume of the tank that is 1372 ft^3. The formula is given by
V = hx^2
We're given that V = 1,372 ft^3, so
hx^2 = 1,372
h = 1,372 / x^2 ----------(equation 1)
Surface area can be calculated using the below given formula.
A = x^2 + 4xh
Substituting the value of euation 1, h in A, we get,
A = x^2 + 4x(1,372 / x^2)
= x^2 + 5,488 / x
To get the minimum surface area to get the minimum weight, we need to differentiate A with respect to x and equate it to zero.
dA/dx = 2x - 5,488 / x^2 = 0
2x^3 - 5,488 = 0
x =\(\sqrt[3]{\frac{5,488}{2} }\)
x= 14
We got the value of x, we substitute the value in equation 1, we get the value of h.
h = 1,372 / 14^2
h= 7
So, the dimensions of the tank with minimum weight are width and length is 14 and height is 7.
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if jonah completes a trip of 240 miles at the rate of 30 mph, at what rate would he have to travel on the return trip in order to average 40 mph for the round trip?
Answers
Answer:
60 mph
Step-by-step explanation:
Each way of the trip is 240 miles.
The round trip is 2 × 240 miles = 480 miles
The first half of the trip, 240 miles, was made at a speed of 30 mph.
s = d/t
st = d
t = d/s
t = 240 miles / 30 mph = 8 hours
The first part of the trip took 8 hours.
The entire round trip is 240 miles × 2 = 480 miles
The speed for the entire trip is 40 mph.
480 miles / 40 mph = 12 hours
12 hours - 8 hours = 4 hours
The return trip takes 4 hours.
The return trip is 240 miles.
s = d/t = 240 miles / 4 hours
s = 60 mph
Answer: 60 mph
- Show BOTH distribution and FOIL to find the product of (3x - 2)and(2x + 6).
Answers
Given the following question:
\((3x-2)\times(2x+6)\)Foil method:
\(\begin{gathered} (3x-2)\times(2x+6) \\ \mleft(3x-2\mright)\mleft(2x+6\mright)=3x\cdot\: 2x+3x\cdot\: 6-2\cdot\: 2x-2\cdot\: 6 \\ 3x\cdot\: 2x+3x\cdot\: 6-2\cdot\: 2x-2\cdot\: 6 \\ 3x\times2x=6x \\ 6\times2=12 \\ 6x+x=7x\times2=14x \\ =6x^2+14x-12 \end{gathered}\)Your answer using the foil method is 6x^2 + 14x - 12
Please help simplify.The variables have no values.
(2/3x³-1/y²)(1/2x²-4/3xy-1/2y²)(y≠0)
Answers
Answer:
\(\dfrac{6x^5y^2-16x^4y^3-6x^3y^4-9x^2+24xy+9y^2}{18y^2}\)
Step-by-step explanation:
The applicable rule of exponents is ...
(a^b)(a^c) = a^(b+c)
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For this problem, it is convenient to factor out a factor that will eliminate fractions. Then the distributive property and the rules for adding fractions apply.
\(\left(\dfrac{2}{3}x^3-\dfrac{1}{y^2}\right)\left(\dfrac{1}{2}x^2-\dfrac{4}{3}xy-\dfrac{1}{2}y^2\right)\qquad\text{given}\\\\=\dfrac{1}{18y^2}(2x^3y^2-3)(3x^2-8xy-3y^2)\\\\=\dfrac{1}{18y^2}(2x^3y^2(3x^2-8xy-3y^2)-3(3x^2-8xy-3y^2))\\\\=\boxed{\dfrac{6x^5y^2-16x^4y^3-6x^3y^4-9x^2+24xy+9y^2}{18y^2}}\)