I need some help with this question​

The expansion of  (z+1)⁶ = z⁶ + 6z⁵+ 15z⁴ + 20z³+ 15z² + 6z + 1

A type of number pattern is Pascal's Triangle. The triangle-shaped arrangement of integers known as Pascal's Triangle provides the coefficients for the expansion of any binomial statement.

The arrangement of the numerals creates a triangle when they are reflected. The number 1 is first placed at the top, and then the numbers are arranged in a triangle design.

The expansion of (x + y)ⁿ is is given by

(x + y)ⁿ = a₀xⁿ + a₁xⁿ⁻¹y + a₂xⁿ⁻²y² + … + a[tex]_{n -1}[/tex] xyⁿ⁻¹ + a[tex]_{n}[/tex]yⁿ  

Here we have (z+1)⁶

Write out Pascal's triangle starting from 1 and up to 6 rows

                                1

                           1         1    

                      1         2         1      

                 1         3         3        1          

            1        4          6         4       1    

      1         5        10         10      5        1  

1          6         15         20      15      6        1    

So coefficients of expansion are 1, 6, 15, 20, 15, 6, and 1  

(z+1)⁶ =  (1)z⁶ + (6)z⁵(1)+ (15)z⁴(1)² + 20z³(1)³ + 15z²(1)⁴ + 6z¹(1)⁵ + (1)z⁰(1)⁶

=> (z+1)⁶ = z⁶ + 6z⁵+ 15z⁴ + 20z³+ 15z² + 6z¹ + 1

Therefore,

The expansion of  (z+1)⁶ = z⁶ + 6z⁵+ 15z⁴ + 20z³+ 15z² + 6z + 1

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Answer:19.28   or 125/7

Step-by-step explanation:

first, replace all the a's with 1's, the b's with 2's, and the c's with 4's. you end up with  (-3(1)+4(2))  ³/2(4)-1

then you would solve that until you get,   (-3+8)³/8-1

then,  (5)³/7

and once you've solved that you get 19.28 or 125/7

If the bases of two exponential expressions are different, they cannot be directly compared or combined.

You can convert one of the bases to match the other by using logarithm properties. For example, if you want to compare or combine 2³ and 3², you can convert the base of one of them to match the other by using the logarithm property log3(3²) = 2, so 3² can be rewritten as 2^(log3(3²)) = 2².

If you want to add or subtract exponential expressions with different bases, you can convert one of the bases to match the other by using logarithm properties, then combine the exponents. For example, if you want to add 2³ and 3², you can convert the base of one of them to match the other by using the logarithm property log2(3) = 1.58496, so 3² can be rewritten as 2^(log2(3)*2) = 2^3.18992. Now you can add the two exponents: 2³ + 2^3.18992 = 2³ (1 + 2^0.18992).

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Answer:

72 divided by 12 is 6 so 6 hours she needs to babysit

The solution to the equation 8.4 = 1.4x for the unknown variable x is 6.

8.4 = 1.4x

divide both sides by the variable of x

x = 8.4 / 1.4

x = 6

Check:

8.4 = 1.4x

8.4 = 1.4(6)

8.4 = 8.4

Using inverse relationship

8.4 = k / 1.4

cross product

8.4 × 1.4 = k

11.76 = k

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The probability that on any one day, Heila will not be late for school is 0.69 or 69%.

We can use conditional probability to find the probability that Heila will not be late for school on any one day.

Let W be the event that Heila walks to school, and L be the event that Heila is late for school.

We know that:

P(W) = 0.65 (the probability that Heila walks to school)

P(L | W) = 0.4 (the probability that Heila is late for school given that he walks to school)

P(L | W') = 0.1 (the probability that Heila is late for school given that he cycles to school)

The probability that Heila is not late for school is the complement of the event that he is late for school. So:

P(L') = 1 - P(L)

We can use the formula for conditional probability to find the probability that Heila is not late for school:

P(L' | W) = 1 - P(L | W) = 1 - 0.4 = 0.6

P(L' | W') = 1 - P(L | W') = 1 - 0.1 = 0.9

We also know that the total probability of the outcomes must be equal to 1. Therefore, we can find the probability that Heila walks to school, P(W) by using the following equation:

P(W) + P(W') = 1

We know that P(W) = 0.65, so we can find P(W') by subtracting P(W) from 1:

P(W') = 1 - P(W) = 1 - 0.65 = 0.35

Now, we can use the formula of Total Probability to find the probability that Heila will not be late for school:

P(L') = P(L' | W) * P(W) + P(L' | W') * P(W')

P(L') = 0.6 * 0.65 + 0.9 * 0.35 = 0.69

Therefore, the probability that on any one day, Heila will not be late for school is 0.69 or 69%.

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The length of t to the nearest centimeter is 1083 cm.

Angles are the figure formed by the intersection of two lines or rays by sharing a common point. This point is called the vertex of the angle.

Given that,

u = 750 cm

m ∠U = 42°

m ∠T = 105°

We have the sine rule to find missing length as,

[tex]\frac{u}{sin U}[/tex] = [tex]\frac{t}{sin T}[/tex]

Substituting the measures given,

[750 / sin (42°)] = [t / sin (105°)]

Cross multiplying,

t = [ 750 × sin (105°)] / sin (42°)

t = 1082.59 ≈ 1083 cm

Hence the length of the missing side is 1083 cm.

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Angela's sample size for the study on predicting how many 7th graders earned an A, is 57 students

The number of observations used to calculate estimates for a certain population is known as the sample size. The sample size was determined by drawing from the population. Sampling is the practice of choosing a portion of the population from which to draw conclusions about the characteristics of the entire population.

In engaging in sampling, Angela wanted to sample 15 % of the 7th graders which means that her sample size would be :

= Percentage to be sampled x Number of 7th graders

= 15 % x 380

= 57 students

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So on solving the provided question, we can say that  y = a + xb is the necessary equation for the total fee the repairman charged, y = 260 + 6x

A mathematical equation is a formula that joins two statements and uses the equal symbol (=) to indicate equality. A mathematical statement that establishes the equality of two mathematical expressions is known as an equation in algebra. For instance, in the equation 3x + 5 = 14, the equal sign places the variables 3x + 5 and 14 apart. The relationship between the two sentences on either side of a letter is described by a mathematical formula. Often, there is only one variable, which also serves as the symbol. for instance, 2x – 4 = 2.

Here,

Let the total amount charged by the repairman be y and the total number of hours worked be x,

[tex]y = a + xb[/tex]

a is as  travel fee

b is as  hourly rate

Therefore, y = a + xb is the necessary equation for the total fee the repairman charged.

y = 260 + 6x

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The solution to the equation 1/h-5+2/h+5=16/h^2-25 is H=7 .

An equation’s solution is an array of all values that, when substituted for unknowns, make the equation true. To solve equations with one unknown raised to a power of one, two basic algebra methods using the additive and multiplicative properties are needed. The set of linear equations has three alternative solutions.

They are as follows:

One-of-a-kind solution.

There is no solution.

There are an infinite number of solutions.

False solutions are ones that do not fundamentally address the problem at hand but fool others into believing they do, while also causing additional major problems.

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Step-by-step explanation:

The question is a bit confusing, so I am going to reword it to what I assume it means, which is, “Sergio initially has “p” paintings. There are 12 art collectors total, including Sergio. All the art collectors together donate a total of 48 paintings, and each art collector donates the same number of paintings. How many paintings does Sergio have left?”

p = number of Sergio’s paintings at the start.

p-X = the number of Sergio’s paintings at the end.

48 = the total number of paintings donated by all art collectors

12 = number of art collectors

48/12 = 4 = the number of paintings that each art collector donated

4 = X = the number of paintings that Sergio donated

p-4 = the number of paintings Sergio has left (answer)

(My regrets if my understanding of the question was something other than the above.)

Answer:

It would be 6 because the exponent is 2 so you multiply 3 twice which equals 6

Step-by-step explanation:

3 to the 2nd exponent is 3^2 = 9.

Answer:

y=-2x+3

Step-by-step explanation:

y=mx+b

Answer:

x = -2.65865384615

Step-by-step explanation:

-91x - 13x = 276.50

First, Combine Like Terms

-91x + (-13x) = -104x

-104x = 276.50

Second, divide

276.50 / -104 = x

x = -2.65865384615

The formula in coordinate geometry for Class 9 include distance formula, midpoint formula, section formula, equation of a line, slope of a line, equation of a circle, equation of a parabola, equation of an ellipse, equation of a hyperbola, area of a triangle, and area of a circle.

The distance formula is used to determine the distance between two points in a coordinate plane. The midpoint formula is used to find the midpoint between two points. The section formula is used to calculate the area of a triangle formed by three points. The equation of a line is used to define the line in terms of its slope and intercept. The slope of a line is used to calculate the angle of the line in relation to the x-axis. The equation of a circle is used to determine the equation of a circle with a given center and radius. The equation of a parabola is used to determine the equation of a parabola with a given focus and directrix. The equation of an ellipse is used to determine the equation of an ellipse with a given center and major and minor axes. In coordinate geometry the equation of a hyperbola is used to determine the equation of a hyperbola with a given center and asymptotes. The area of a triangle is used to find the area of a triangle given the three vertices. The area of a circle is used to find the area of a circle given the radius.

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The least whole number possibility for the third side is 8 cm. The greatest whole number possibility for the third side is 22 cm.

A triangle is a three-sided polygon. It is a geometric shape that has three edges and three vertices. Triangles can be classified based on their side lengths (such as equilateral, isosceles, or scalene) or based on the angles between their sides (such as acute, right, or obtuse). Triangles are a fundamental shape in geometry and are used in many branches of mathematics and physics. To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Using this rule, we can find the possible range of values for the third side of the triangle.

The least possible length of the third side would be:

8.5cm + 15cm - 15cm = 8.5cm

The greatest possible length of the third side would be:

(8.5cm + 15cm) - 1cm = 22.5cm

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Lilyan took out a 72-month car loan for $17,595. If he had paid a total of $22,556. 79 at the end of the loan, then the interet rate is 6.7%

What is principal amount?

The loan amount that the lender gives to the borrower is referred to as the primary amount. Additionally, it is the factor the lender uses to calculate interest.

What is simple interest?

Simple interest is a method to calculate the amount of interest charged on a sum at a given rate and for a given period of time. The formula to calculate simple interest is  I=P.r.t

Where P=principal amount, r=rate, t=time.

Lilyan took out a 72-month car loan for $17,595.

If he had paid a total of $22,556. 79 at the end of the loan,

find the interet rate

72 months = 6 years

principal = 17595

amount = 22556.79

we have to calculate interest

A = P ( 1+ r/100) ^ n

 = 17595 ( 1+r/100)^6

22556.79/17595 ^ 1/6= 1+0.00r

r= 6.7%

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The principal should use Square meters because the average length and width of the parking spaces will determine the number of parking spaces created in the rectangular parking lot.

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By graphing a plot point using the x-intercept form, the function (x + 4)2 is shown to be x2 + 16 + 8x.

The subject of mathematics includes quantities and their variations, equations and related structures, forms and their locations, and places where they can be found. The term "function" refers to the relationship between a set of inputs, each of which has an associated output. A relationship between inputs and outputs in which each input leads to a single, distinct result is known as a function. Usually, f is used to denote functions (x). input is an x. There are four main types of functions accessible. based on the following factors: on functions, one-to-one functions, many-to-one functions, within functions, and on functions.

given

(x + 4)²

= x2 + 16 + 8x

By graphing a plot point using the x-intercept form, the function (x + 4)2 is shown to be x2 + 16 + 8x.

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The coordinates of A'', B'' and C'' are (-1, 4), (-4, 3) and (-1, 2) respectively.

Transformation of geometrical figures or points is the manipulation of a given figure to some other way.

Different types of transformations are Rotation, Reflection, Glide reflection, Translation and Dilation.

Given triangle has coordinates,

A(2, -2), B(5, -3) and C(2, -4).

First the triangle is translated by the rule (x, y) → (x - 1, y + 6)

A(2, -2) becomes A'(2 - 1, -2 + 6) = A'(1, 4).

B(5, -3) becomes B'(5 - 1, -3 + 6) = B'(4, 3)

C(2, -4) becomes C'(2 - 1, -4 + 6) = C'(1, 2)

Then the translated triangle is reflected across the Y axis.

When reflected a point (x, y) across the Y axis, y coordinate remains same and x coordinate flips.

A'(1, 4) becomes A''(-1, 4)

B'(4, 3) becomes B''(-4, 3)

C'(1, 2) becomes C''(-1, 2)

Hence the vertices of the triangle after undergoes translation and reflection becomes A''(-1, 4), B''(-4, 3) and C''(-1, 2).

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Answer: 951 ft

Step-by-step explanation:

-230 + 602 - 183 + 762
= 951 ft

The correct equation of step 2 is 16x = 14

from the question, we have the following parameters that can be used in our computation:

14 + 12x - 6 = 18 - 4x + 4

When the like terms on each side are Evaluated, we have

Step 1: 8 + 12x = 22 - 4x

When the like terms on all sides are Evaluated, we have

Step 2: 16x = 14

Hence, the correct equation is 16x = 14

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So on provided question we cans ay that Take the common binomial into account. Note that when we multiply our response by itself, the original polynomial results.

A mathematical expression made up of coefficients and uncertainty that only makes use of additions, subtractions, multiplications, and positive integer powers of variables is known as a polynomial. The formula x2 4x + 7 denotes a single indeterminate x polynomial. An expression in mathematics known as a polynomial is made up of variables (also known as indeterminates) and coefficients that may be added, subtracted, multiplied, and raised to negative integer powers of non-variables. An algebraic statement with variables and coefficients is called a polynomial. An expression can only contain the operations addition, subtraction, multiplication, and non-negative integer exponents. Polynomials are the name for these expressions.

here,

Group the first two terms first, followed by the last two terms. Remove a GCF from each each binomial. Take the common binomial into account. Note that when we multiply our response by itself, the original polynomial results.

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Therefore , the solution to the given problem of equation comes out to be is 19 mph for a boat in calm water.

An equation must always include two equal members and one or more unknown members (the unknown variables). When multiplied by itself, what number, for example, results in 16. To express this maths equation mathematically, use the equation or expression.

Here,

The determining formula is

30/u-4 = 46/u+4

(time equation, where u represents the boat's speed in still water)

Cross-multiply:

30*(u+4) = 46(u-4) (u-4),

30u + 120 = 46u - 184,

120 + 184 = 46u - 30u,

Thus,

304 = 16u

=> 304/16

=> 19.

The answer is 19 mph for a boat in calm water.

Therefore , the solution to the given problem of equation comes out to be is 19 mph for a boat in calm water.

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The following actions must be taken in order to simplify the rational function f (x) = p (x) q (x): Where q (x) = 0, identify the values of x.

The study of mathematics include the study of numbers, formulas and related structures, forms and the environments in which they exist, quantities and their variations, and the environments in which they exist. A function is an association between a number of inputs, each of which has an output. Simply described, a function is a relationship between inputs and outputs in which each input leads to a single, distinct result.  Usually, the letter f is used to denote functions (x). input consists of x. There are four different types of functions. based on the following elements: one-to-one functions, many-to-one functions, on functions, one-to-one functions, and within functions.

given

The following actions must be taken in order to simplify the rational function f (x) = p (x) q (x): Where q (x) = 0, identify the values of x.

If so, all real values, with the exception of these roots, are in the domain of f (x).

It may be necessary to factor by grouping or by applying the factor theorem in order to fully factor both p (x) and q (x).

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(a) The exponential growth model is P = 5655 e^(0.014t)

(b) Population of the town in 2022 is 6689.5065.

Population growth is defined as the number of people increasing over years in a specific region calculated using annual growth rate.

(a) The formula to calculate population growth is,

P = P₀ e^(r t)

Given,

Initial population in 2010, P₀ = 5655

Annual growth rate, r = 1.4% = 0.014

Substituting,

P = 5655 e^(0.014t)

(b) To estimate the population of town in 2022,

t = 2022 - 2010 = 12 years

Substituting value of t in the formula in (a),

P = 5655 e^(0.014 × 12)

  = 5655 e^(0.168)

  = 5655 × 1.1829

  = 6689.5065

Hence the population is 6689.5065 in 2022.

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So, on solving the provided question, we can say that coefficient of y² in the expression 3y² -  4x is 3

A coefficient in mathematics is a polynomial, a series, or the multiplicative coefficient of a particular term in an expression. Typically numeric, however any expression is permitted. The term "parameter" can also refer to the coefficients themselves if they are variables. A number times a variable equals a coefficient. Coefficient examples include:  The coefficient is 14 in phrase 14c 14c 14c. The coefficient is 1 for word g. multiplies the variable by this amount. example  Given that "z" is a variable and that 6z is the definition of the term, the coefficient is 6. One is the coefficient of the square of x.

here,

coefficient of y² in the expression 3y² -  4x is 3

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The value for constant of proportionality and inverse proportionality is k=12 and x=12/y respectively.

What is constant of proportionality and inverse proportionality?

The ratio between two proportional quantities' constant values is known as the constant of proportionality. When either their ratio or their product gives a constant, two changing values are said to be in a proportionate relationship.

When the value of one item rises in relation to a fall in the other or vice versa, two values are said to be inversely proportionate. This indicates that the natural behaviour of these two quantities is contrary

In a bakery workers must make 12 batches of bagel dough an hour.

The number of batches of dough, y, that each worker needs to mix in one hour is inversely proportional to the number of workers, x.

Lets find the constant of proportionality first.

According to the graph -

The point given in the graph is (6,2).

x=6 and y=2

(where 'x' denotes  no of workers and 'y' denote no. of bagel dough batches).

Equation for constant of proportionality is -

x×y=k

6×2=k

k=12 is the constant of proportionality.

Now find the equation for inverse proportionality.

No. of workers (along x-axis) = x

No. of the bagel dough batches (along y-axis) = y

The algebraic equation relating both = x×y=k

The constant of proportionality is k=12.

x×y=12

xy=12

x=12/y

Therefore, the equation of inverse proportionality is x=12/y.

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