Prove that if x is a nontrivial square root of 1, modulo n, then gcd(r- 1, n) and ged(x + 1.n) are both nontrivial divisors of n.

The expression cos (125°) - cos 55° cos 35° cos 55° can be written as cos (55°) + cos (55°) cos (35°) cos (55°).

cos (125°) can be rewritten as cos (180° - 125°). Similarly, cos (35°) can be rewritten as cos (180° - 35°). Therefore, the expression can be written as:

cos (180° - 125°) - cos (55°) cos (180° - 35°) cos (55°)

Simplifying further, we have:

cos (55°) - cos (55°) cos (145°) cos (55°)

Since 145° is the supplement of 35°, we can rewrite it as:

cos (55°) - cos (55°) cos (180° - 35°) cos (55°)

Now, cos (180° - 35°) is equal to -cos (35°). Therefore, the expression becomes:

cos (55°) + cos (55°) cos (35°) cos (55°)

Hence, the expression as a function of an acute angle is:

cos (55°) + cos (55°) cos (35°) cos (55°)

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Answer:

B. No

Step-by-step explanation:

To determine if the point (-8, -2) is a solution to the equation y = 2, we need to check if the y-coordinate of the point is equal to 2.

In this case, the y-coordinate of the point (-8, -2) is -2, which is not equal to 2. Therefore, (-8, -2) is not a solution to the equation y = 2.

So the answer is B. No.

Answer: no

Step-by-step explanation:y=2 would just be a horizontal line across the 2 in the y-axis. But (8,2) would be a solution.

Answer:

A

Step-by-step explanation:

The question is actually: 2 \sqrt[3]{x^3} ·\sqrt{16x}

The expression is equivalent to option A, \(\rm 8x \sqrt[6]{x}\).

An expression is a mathematical statement that consists of variables, constants, and mathematical operators.

The expression is \(\rm 2\sqrt[3]{x^2} . \sqrt{16x}\)

The expression can be simplified as

\(\rm = 2 . 4 x^{2/3} x^{1/2}\\\\\=8 x ^{ 7/6}\\\\\\ = 8x x^{1/6}\\\\= 8x \sqrt[6]{x}\)

Therefore, the expression is equivalent to option A, \(\rm 8x \sqrt[6]{x}\).

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Therefore, using algebra, Laura is 14 years old and Juan is 21 years old.

Mathematical symbols and the rules for manipulating them are the focus of the discipline of study known as algebra. In order to express unknowable quantities, it makes use of symbols and variables as well as the study of mathematical equations and their characteristics. In algebra, numbers and mathematical operations are represented by letters and symbols in mathematical expressions and equations.

Let x be Laura's age and y be Juan's age.

Use the given information to set up two equations:

y = x + 7 (Juan is 7 years older than Laura)

xy = 294 (The product of their ages is 294 years)

Substitute the first equation into the second equation to get:

x(x + 7) = 294

Simplify the equation by expanding the left side:

x² + 7x = 294

Move all the terms to the left side of the equation:

x² + 7x - 294 = 0

Solve for x by factoring the left side of the equation or using the quadratic formula. Here, we'll use factoring:

(x + 21)(x - 14) = 0

This gives us two possible solutions for x: x = -21 or x = 14.

Since we're looking for age, we can discard the negative solution, x = -21.

Therefore, Laura's age is x = 14.

To find Juan's age, use the first equation and substitute the value of x:

y = x + 7

y = 14 + 7

y = 21

Therefore, Juan's age is y = 21.

So the ages we're talking about are Laura being 14 years old and Juan 21 years old.

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Your question is in Spanish. The English translation is:

Juan is 7 years older than Laura. The product of their ages is 294 years. What age numbers are we talking about?

Answer:

Step-by-step explanation:

So the amount after 4 yrs = amount after 3 yrs plus interest for 1 year;

one year =854-815=39 rate of simple interest;

so 3 yrs x 39 simple interest = 117;

so 815-117 = 698

the sum is 698

22084.78 gallons is the volume of your pool.

Any rectangular solid's volume, V, is equal to the sum of its dimensions, height, width, and depth. In terms of the area of the base, we might alternatively express the formula for a rectangular solid's volume. The base's surface area, B, is equal to its length x width.

Dimensions of the pool = 4.93 m x 9.69 m x 1.75 m

Volume = 4.93 x 9.69 x 1.75

             = 83.6 cubic meter

1 cubic meter = 264.172 gallons

83.6 cubic meter = 83.6 x 264.172

                            = 22084.78 gallons

Hence, 22084.78 gallons is the volume of your pool.

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Answer:

M=1

Step-by-step explanation:

7M-1=6

7M=6+1

7M=7

M=7/7

M=1

Answer: \(x=-2 , y = 6\)

Step-by-step explanation: Use substitution to solve this system because the x value for y is given. Plug it into the second equation.

\(x+(-3x) = 4\)

Then add like terms,

\(-2x = 4\)

Which means,

\(x=-2\)

Plug it into the first equation to solve for y

\(y= -3(-2)\)

Which means,

\(y=6\)

The area of the base of the bird feeder is 49 square inches.

What is area of square?

The area of a square is the amount of space inside the square, and it is calculated by multiplying the length of one side by itself.

To find the area of the base of the bird feeder, we need to find the areas of the triangle and the square and then add them together.

Area of the triangle = (base x height) / 2

= (7 x 4) / 2

= 14 square inches

Area of the square = length x width

= 5 x 7

= 35 square inches

Total area of the base = area of triangle + area of square

= 14 + 35

= 49 square inches

Therefore, the area of the base of the bird feeder is 49 square inches.

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Answer:

Helen has $35 and Brian has $7

Step-by-step explanation:

5b + b = 42

6b = 42

/6 /6

b = 7

7(5) + 7 = 42

35 + 7 = 42

42=42

6 or more terms should be used to estimate the sum of the entire series with an error of less than 0.0001.

To determine the number of terms needed to estimate the sum of the series within an error of less than 0.0001, we can apply the Alternating Series Estimation Theorem. The given series is (-1)^n * 1 / (n + 15)^4. Let's break down the steps to find the required number of terms:

The Alternating Series Estimation Theorem states that if a series is alternating, meaning its terms alternate in sign, and the absolute value of each term is decreasing, then the error made by approximating the sum of the series with a partial sum can be bounded by the absolute value of the first omitted term.

We want to estimate the sum of the series with an error of less than 0.0001. This means that we need to find the number of terms such that the absolute value of the first omitted term is less than 0.0001.

The given series is an alternating series as it alternates in sign with (-1)^n. To ensure the series satisfies the conditions of the Alternating Series Estimation Theorem, we need to verify that the absolute value of each term is decreasing.

Let's examine the absolute value of each term: |(-1)^n * 1 / (n + 15)^4|. Since the numerator is always 1 and the denominator is (n + 15)^4, we can see that the absolute value of each term is indeed decreasing as n increases.

Now, we need to find the number of terms such that the absolute value of the first omitted term is less than 0.0001. Let's denote this number of terms as N.

We can set up an inequality based on the first omitted term: |(-1)^(N+1) * 1 / (N + 15)^4| < 0.0001.

To simplify the inequality, we can remove the absolute value signs and solve for N:

(-1)^(N+1) * 1 / (N + 15)^4 < 0.0001.

Considering the (-1)^(N+1) term, we know that its value alternates between -1 and 1 as N increases. Therefore, we can ignore it for now and focus on the other part of the inequality:

1 / (N + 15)^4 < 0.0001.

To eliminate the fraction, we can take the reciprocal of both sides:

(N + 15)^4 > 10000.

Taking the fourth root of both sides, we have:

N + 15 > 10.

Solving for N, we get:

N > 10 - 15,

N > -5.

Since the number of terms must be a positive integer, we can round up to the nearest whole number:

N ≥ 6.

Therefore, 6 or more terms should be used to estimate the sum of the entire series with an error of less than 0.0001.

In summary, according to the Alternating Series Estimation Theorem, 6 or more terms should be used to estimate the sum of the series (-1)^n * 1 / (n + 15)^4 with an error of less than 0.0001.

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To calculate the volume of a 1.420M NaOH solution required to titrate 25mL of a 4.500M H2SO4 solution, we can use the stoichiometry of the balanced chemical equation and the concept of molarity.

The balanced chemical equation for the reaction between NaOH and H2SO4 is:

2NaOH + H2SO4 -> Na2SO4 + 2H2O

From the equation, we can see that 2 moles of NaOH react with 1 mole of H2SO4 to form 1 mole of Na2SO4 and 2 moles of H2O.

To find the volume of the NaOH solution required, we can use the molar ratios between NaOH and H2SO4. Since the molarity of the H2SO4 solution is given as 4.500M, it represents the number of moles of H2SO4 per liter.

By using the molar ratios from the balanced equation, we can calculate the moles of NaOH required to react with the given moles of H2SO4. Then, by dividing the moles of NaOH by the molarity of the NaOH solution, we can find the volume in liters. Finally, we convert the volume to milliliters.

Please provide the volume of the H2SO4 solution used in milliliters, as it is needed to perform the calculation.

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The robot moves in the positive direction along a straight line so that after t minutes its distance is s=6t^4 feet from the origin. (a) Find the average velocity of the robot over intervals 2, 4. We have the following data: Initial time, t₁ = 2 min.

Final time, t₂ = 4 min.The distance from the origin is given by s = 6t^4Therefore, s₁ = s(2) = 6(2^4) = 6(16) = 96 feet s₂ = s(4) = 6(4^4) = 6(256) = 1536 feet

We can find the average velocity of the robot over the interval 2, 4 as follows: Average velocity = (s₂ - s₁) / (t₂ - t₁)Average velocity = (1536 - 96) / (4 - 2)Average velocity = 1440 / 2Average velocity = 720 feet per minute(b) Find the instantaneous velocity at t=2.To find the instantaneous velocity at t = 2 min, we need to take the derivative of the distance function with respect to time. We have the distance function as:s = 6t^4 Taking derivative of s with respect to t gives the velocity function:v = ds / dt Therefore,v = 24t³At t = 2, the instantaneous velocity is:v(2) = 24(2)³v(2) = 24(8)v(2) = 192 feet per minute Therefore, the instantaneous velocity at t = 2 min is 192 feet per minute.

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2,5,6 2)AB and MN 5)B and N 6)CB and LN

Step-by-step explanation:

Edge 2021

The graph of ƒ has a point of inflection  is (D) -3.259, 0, and 1.603

To determine if a function has a point of inflection, we need to find the values of x where the second derivative changes sign from positive to negative or vice versa. The sign of the second derivative at a point determines the concavity of the function at that point. If the second derivative is positive, the function is concave up, and if it is negative, the function is concave down.

Since ƒ"(a)=x² cos(2a+22), we can see that the second derivative is positive for x²>0. So the function is concave up when x²>0, which means when x is in the interval (-∞,0)∪(0,∞).

Since the interval (-4,3) includes both negative and positive values of x, the graph of ƒ may have a point of inflection within this interval. To determine the exact points of inflection, we need to find the values of x where the second derivative equals zero or is undefined.

Therefore, the answer is (D) -3.259, 0, and 1.603.

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The domain is the input value of x, which can be any real number.

There is no inequality.

Set notation x | E R. ( all real nUmbers)

Interval. Oration (-infinity, infinity)

Range: set x to 0 and solve:

3(0)^2 + 2 = 2

The lowest value is 2 and goes to infinity.

Inequality is >=

Set notation Y | >= 2

Interval notation : [2, infinity)

End behavior they both go to infinity .

Answer:

25.65634 squares cm

Step-by-step explanation:

The area of the whole circle is

pi x 7 squared

pix49 =153.93804

the degrees of a circle are 360

 60 is 1/6 of 360

so we need to divide the area by 6

153.93804/6 = 25.65634

Answer:

124

Step-by-step explanation:

The sum of interior angles in a triangle is equal to 180:

m<R + m<S + m<T = 180

11x + 3 + x + 15 + 2x + 8 = 180 add like terms

14x + 26 = 180 subtract 26 from both sides

14x = 154 divide both sides by 14

x = 11 to find m<R replace x with 11

11*11 + 3 = 124

Apply angle sum property

\(\\ \rm\rightarrowtail 11x+3+x+15+2x+8=180\)

\(\\ \rm\rightarrowtail 14x+26=180\)

\(\\ \rm\rightarrowtail 14x=154\)

\(\\ \rm\rightarrowtail x=11\)

There are 144 combinations of 1 entree, 2 vegetables, and 1 dessert that can be selected from 4 entrees, 6 vegetables, and 6 desserts.

To determine the number of combinations, we multiply the number of options for each category.

For the entree, we have 4 options to choose from.

For the vegetables, we need to select 2 out of 6, which can be done in 6 choose 2 ways.

This is calculated as 6! / (2!(6-2)!), which simplifies to

6! / (2!4!)

Similarly, for the dessert, we have 6 options to choose from.

To calculate 6 choose 2, we can use the formula for combinations:

n choose r = n! / (r!(n-r)!).

Plugging in the values, we have

6! / (2!4!) = (6 × 5 × 4 × 3 × 2 × 1) / [(2 × 1) × (4 × 3 × 2 × 1)] = 15.

Therefore, we have 4 options for the entree, 15 options for the vegetables, and 6 options for the dessert.

Multiplying these numbers together, we get 4 × 15 × 6 = 144.

Therefore, there are 144 possible combinations of 1 entree, 2 vegetables, and 1 dessert, given the options of 4 entrees, 6 vegetables, and 6 desserts.

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Answer:

does it give the hight of the truck??

Step-by-step explanation:

Answer:

1005.3 ft³

Step-by-step explanation:

cylinder has a base diameter of 8ft and a height of 20ft. What is it's volume in cubic ft, to the nearest 10th place?

Given that:

Base diameter = 8feets

Radius, r = 8/2 = 4 feets

Height h, = 20 feets

Volume of cylinder, V = πr²h

V = π * 4² * 20

V = π * 16 * 20

V = 1005.3096

V = 1005.3096 ft³

Answer:

A

Step-by-step explanation:

Range = Highest value - Lowest value

At Park Zoo:

R = 44 - 38

= 6

At Countryside Zoo;

R = 45 - 38

= 7

So, it cannot be D, because the ranges are not the same and it cannot be B because the range is greater at countryside zoo

The mode is the value that appears the moat frequently.

At Park zoo, the mode is 41 since it has the most dots.

At Countryside zoo, the mode is 40, since it has the most dots

So, it cannot be C because Park zoo has a greater Mode.

so, the only answer is A

Answer:

Step-by-step explanation: whats this

There are 2,300 possible 5-topping pizzas that can be made with 14 different toppings and no double-orders of toppings allowed.

If the pizza parlor offers 14 different toppings and no double-orders of toppings are allowed, the number of 5-topping pizzas possible can be calculated using the combination formula:

nCr = n! / (r! × (n-r)!)

where n is the total number of items to choose from (14 toppings in this case) and r is the number of items to be selected (5 toppings for a pizza).

Therefore, the number of 5-topping pizzas possible can be calculated as:

14C5 = 14! / (5! × (14-5)!)

= (14 × 13 × 12 × 11 × 10) / (5 × 4 × 3 × 2 × 1)

= 2002

Therefore, there are 2002 possible 5-topping pizzas that can be ordered from the pizza parlor.

To calculate the number of 5-topping pizzas possible when there are 14 different toppings available and no double-orders of toppings are allowed, we can use the formula for combinations, which is:

n C r = n! / (r! × (n-r)!)

where n is the total number of items, r is the number of items being selected, and ! denotes the factorial operation.

In this case, we have:

n = 14 (the total number of toppings)

r = 5 (the number of toppings being selected)

Plugging these values into the formula, we get:

14 C 5 = 14! / (5! × (14-5)!)

= (14 × 13 × 12 × 11 × 10) / (5 × 4 × 3 × 2 × 1)

= 2,300

To calculate the number of possible 5-topping pizzas, we need to use the combination formula since the order of the toppings doesn't matter. The formula is:

n C r = n! / (r! × (n-r)!)

where n is the total number of items to choose from, r is the number of items to choose, and "!" denotes the factorial function (i.e., the product of all positive integers up to that number).

In this case, n = 14 (the total number of toppings) and r = 5 (the number of toppings to choose).

So, the number of possible 5-topping pizzas is:

14 C 5 = 14! / (5! × (14-5)!)

= (14 × 13 × 12 × 11 × 10) / (5 × 4 × 3 × 2 × 1)

= 2,002,200

Therefore, there are 2,300 possible 5-topping pizzas that can be made with 14 different toppings and no double-orders of toppings allowed.

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Coordinates of QRST:

Rule for a reflection accros the y-axis:

Find the coordiantes of the figure after the reflection across y-axis:

A rotation 360º clockwise is a full rotation, the figure still in the same direction after the rotation (the coordinates of the points don't change)

Answer:

(1, 2); (6, 7)

Step-by-step explanation:

Step-by-step explanation:

refer to the picture.. hope this helps :)

Answer: 63 Stickers

Step-by-step explanation:

Given information:

Ratio = Slade : Corbett = 5 : 2

Corbett has 27 fewer stickers

Set variables:

Let x be the number of stickers Corbett has

Let x + 27 be the number of stickers Slade has

Set proportional equation:

\(\frac{2}{5}~ =~\frac{x}{x~+~27}\)

Cross multiply the system

\(2~(x~+~27)~=~5~*~x\)

Simplify by distributive property

\(2~*~x~+~2~*~27~=~5x\)

\(2x~+~54~=~5x\)

Subtract 2x on both sides

\(2x~+~54~-~2x~=~5x~-~2x\)

\(54~=~3x\)

Divide 3 on both sides

\(54~/~3~=~3x~/~3\)

\({x=18}\)

Add Corbett's and Slade's amounts together

Corbett = x = 18 stickers

Slade = x + 27 = 18 + 27 = 45 stickers

Total = 18 + 45 = \(\Large\boxed{63~Stickers}\)

Hope this helps!! :)

Please let me know if you have any questions

Answer:

63 stickers

Step-by-step explanation:

Define the variables:

Given ratio:

Slade : Corbett = 5 : 2

Substitute the defined variables:

\(\implies \sf x : x - 27 = 5 : 2\)

\(\implies \sf \dfrac{x}{x-27}=\dfrac{5}{2}\)

Cross multiply:

\(\implies \sf 2x=5(x-27)\)

Expand:

\(\implies \sf 2x=5x-135\)

Subtract 5x from both sides:

\(\implies \sf -3x=-135\)

Multiply both sides by -1:

\(\implies \sf 3x=135\)

Divide both sides by 3:

\(\implies \sf x=45\)

Therefore, Slade had 45 stickers.

Substitute the found value of x into the expression for the number of stickers Corbett had:

\(\implies \sf 45-27=18\)

Therefore, Corbett had 18 stickers.

Total number of stickers = 45 + 18 = 63

To calculate the discharge through the circular channel, we can use Manning's equation, which relates the flow rate (Q) to the channel properties and flow conditions. Manning's equation is given by:

Q = (1/n) * A * R^(2/3) * S^(1/2)

where:

Q is the discharge (flow rate)

n is Manning's coefficient (0.014 in this case)

A is the cross-sectional area of the channel

R is the hydraulic radius of the channel

S is the slope of the channel bed

First, let's calculate the cross-sectional area (A) of the circular channel. The diameter of the channel is given as 6m, so the radius (r) is half of that, which is 3m. Therefore, the area can be calculated as:

A = π * r^2 = π * (3m)^2 = 9π m^2

Next, let's calculate the hydraulic radius (R) of the channel. For a circular channel, the hydraulic radius is equal to half of the diameter, which is:

R = r = 3m

Now, we can calculate the slope (S) of the channel bed. The given slope is 1 in 600, which means for every 600 units of horizontal distance, there is a 1-unit change in vertical distance. Therefore, the slope can be expressed as:

S = 1/600

Finally, we can substitute these values into Manning's equation to calculate the discharge (Q):

Q = (1/0.014) * (9π m^2) * (3m)^(2/3) * (1/600)^(1/2)

Using a calculator, the discharge can be evaluated to get the final result.

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Answer:

50,000 in scientific notation is 5 x 10^4

Step-by-step explanation:

10^4 = 10,000 x 5 = 50,000