Charlie and Kathy want to borrow $20,000 to make some home improvements. Their bank will lend them the money for 10 years at an interest rate of 5.75 %. How much will they pay in interest? (Round monthly payment calculations to 2 decimal places.)

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. (x+4)⁸ is expanded as x⁸ + 32x⁷ + 448x⁶ + 3584x⁵ + 17920x⁴ + 57344x³ + 114688x² + 131072x + 65536

The Binomial theorem tells us how to expand expressions of the form (a + b)ⁿ, for example, (x + y)⁷. The larger the power is, the harder it is to expand expressions like this directly.

The total number of terms in the expansion of (x+y)ⁿ are (n+1).

The sum of exponents of x and y is always n.

nC₀, nC₁, nC₂,….., nCₙ are called binomial coefficients and also represented by C₀, C₁, C₂, ….., Cₙ

The binomial coefficients which are equidistant from the beginning and from the ending are equal.

i.e. nC₀ = nCₙ, nC₁ = nCₙ₋₁ , nC₂ = nCₙ₋₂ ,….. etc.

Let the expression be (x + 4)⁸

Use the binomial expansion theorem to find each term. The binomial theorem states

(a+b)n=n∑k=0nCk⋅(an−kbk)(a+b)n=∑k=0n⁡nCk⋅(an-kbk).

8∑k=08!(8−k)!k!⋅(x)8−k⋅(4)k∑k=08⁡8!(8-k)!k!⋅(x)8-k⋅(4)k

Expand the summation

8!/(8−0)!0!.(x)⁸⁻⁰⋅(4)⁰+8!/(8−1)!1!(x)⁸⁻¹⋅(4)¹+…+8!/(8−8)!8!(x)⁸⁻⁸⋅(4)⁸

Simplify the exponents for each term of the expansion.

1⋅(x)⁸⋅(4)⁰+8⋅(x)⁷⋅(4)1+…+1⋅(x)⁰⋅(4)⁸¹⋅(x)⁸⋅(4)⁰+8⋅(x)⁷⋅(4)¹+…+1⋅(x)⁰⋅(4)⁸

Simplify the polynomial equation, we get

x⁸ + 32x⁷ + 448x⁶ + 3584x⁵ + 17920x⁴ + 57344x³ + 114688x² + 131072x + 65536

Therefore, by expanding the binomial equation, we get

(x + 4)⁸ =  x⁸ + 32x⁷ + 448x⁶ + 3584x⁵ + 17920x⁴ + 57344x³ + 114688x² + 131072x + 65536

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Answer:

[tex]10j \leqslant 130[/tex]

[tex]j \leqslant 13[/tex]

Solution of this polynomial equation is y = 12

In mathematics, simply polynomial or polynomial equation can be defined as the equation which consist of coefficient, variables and exponents. A polynomial equation has positive numbers as their exponent which is also known as its degree. All the equations such as quadratic equation, cubic equation which has non  negative number as their powers are part of polynomial equation.

Types of polynomial equations :

Some examples of polynomial equation are :

According to question,

2y+1=25

=> 2y +1 -1 = 25-1

=> 2y = 24

=> y = 12

Thus solution for given equation is y=12.

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3.22 approx. is the value of x in equation [tex]e^{\frac{x}{2}[/tex] = 5 solved by using natural logarithms

We have been asked to find x in [tex]e^{\frac{x}{2}[/tex] = 5 using natural logarithms

⇒  [tex]e^{\frac{x}{2}[/tex] = 5

⇒ [tex]\frac{x}{2}[/tex] = ㏑(5)

⇒ x = ㏑(25)

      = 3.22 approx.

A number's natural logarithm is its logarithm to the base of the transcendental and irrational number e, which is roughly equivalent to 2.718281828459.

Typically, the natural logarithm of x is denoted by the symbols ln x, loge x, or, if the base e is implicit, just log x. When adding parentheses for clarity, the values ln(x), loge(x), or log (x). In order to avoid ambiguity, this is notably done when the argument to the logarithm is not a single symbol.

The power to which e would need to be raised in order to equal x is the natural logarithm of x.

For instance, e2.0149... = 7.5, thus ln 7.5 equals 2.0149.

Since e1 = e, the natural logarithm of e itself, or ln e, is equal to 1, while the natural logarithm of 1 is 0, since e0 = 1.

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Use natural logarithms to solve each equation.

[tex]e^{\frac{x}{2}[/tex] = 5

Answer:

2

Step-by-step explanation:

(0,2) is the y-intercept

(a) Grade = 2.86%

(b) Grade = 5.85%

Find the road grade to two decimal places:

Grade in terms of percentage = (rise/run) * 100%

(a) Grade = (40/1400) * 100% = 2.86%

(b) Grade = 5.85%

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Answer:

Step-by-step explanation:

hey Can u help me answer this question ? The way you use your money reveals your true priorities. Knowing that, how could having budget categories help you prioritize your money ?

The solution for the radical equation √(x+3) = 4√x-2 is 7 / 3.

The phrase "algebraic expression" refers to any arrangement of terms that have undergone operations like addition, subtraction, multiplication, division, etc. (or variable expression).. Let's use the equation 5x + 7 as an example. Thus, 5x + 7 is an illustration of an algebraic expression.

Eliminate the radical on the both sides by squaring on both sides- 

Both sides must be raised to the second power.

√(x+3) ²=(4√x-2 )²

After squaring we get

x+3=16(x-2)

Solving the equation-

We get rearranged equation as :

x+3=16x-32

15x=35 x=35/15 which further reduces to 7/3 .

Check if the  solution is correct by putting it in original equation-

putting x=7/3 in the equation √(x+3) = 4√x-2 

we get √16/3=√16/3

Therefore, the solution for the radical equation √(x+3) = 4√x-2 is 7 / 3.

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Answer:

3x^3 -8*x^2 + 8*x + 3

Step-by-step explanation:

The standard form for a plolynomial is:

a*x^n + b*x^(n-1) + c*x^(n - 2) ........

We have:

(x-1)(3x^2-5x-3)

this is equal to:

3*x^3 - 5*x^2 + 3*x - 3*x^2 +5*x + 3

where the thing that i did is distribute the multiplication

this is equal to:

3x^3 -8*x^2 + 8*x + 3

The domain of the function is (0,∞) , {x|x>0}

and the range is  (₋∞, ∞) ,  {y|y ∈ R}

Given the function is y = log₅x

A function's domain and range are its constituent parts. A function's range is its potential output, whereas its domain is the set of all possible input values. Range, Domain, and Function. A is the domain and B is the co-domain if a function f: A B exists that maps every element of A to an element in B. 'b', where (a,b) R, provides the representation of an element 'a' under a relation R. The set of images is the function's range. In general, a function's domain and range are denoted as follows: Range(f) = f(x): x domain(f) and domain(f) = f(x): x R

hence we get the domain and range as (0,∞) , {x|x>0} and  (₋∞, ∞) ,  {y|y ∈ R} respectively.

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Based on the given function on the number of pears on the pear tree, the true statement is that The number of pears on the tree increases by 2 each week.

A linear regression line takes the form:

y = mx + b

y = number of pears

m = change in the number of pears per week

x = number of weeks

b = number of pears at first week

When the function is converted to a linear regression form, it takes the form:

P(x) = 2(2+x)

P(x) = 4 + 2x

This shows that the change in the number of pears per week is 2. This means that the number of pears on the tree increases by 2 each week

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keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the PQ line

[tex]P(\stackrel{x_1}{-3}~,~\stackrel{y_1}{9})\qquad Q(\stackrel{x_2}{8}~,~\stackrel{y_2}{-2}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{-2}-\stackrel{y1}{9}}}{\underset{run} {\underset{x_2}{8}-\underset{x_1}{(-3)}}} \implies \cfrac{-11}{8 +3}\implies -1[/tex]

if indeed RS || PQ, then their slopes are exactly the same, thus

[tex]R(\stackrel{x_1}{-8}~,~\stackrel{y_1}{8})\qquad S(\stackrel{x_2}{0}~,~\stackrel{y_2}{y}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{y}-\stackrel{y1}{8}}}{\underset{run} {\underset{x_2}{0}-\underset{x_1}{(-8)}}} \implies \cfrac{y -8}{0 +8}~~ = ~~\stackrel{\stackrel{slope}{\downarrow }}{-1} \\\\\\ \cfrac{y-8}{8}=-1\implies y-8=-8\implies \boxed{y=0}[/tex]

The solutions to the equations are:

A. b = ac² (dark green); b = 2

B. v = m - k (light blue); v = 5

C. p/q = r/s; q = 8

D. m = 3n/4 (red); m = 6

E. x = y(w + z) (light green); x = 10

F. d1 = 2m - d2 (dark blue); d1 = 7

G. s = 8r/9 (purple); s = 4

H. w = [tex]\sqrt{\frac{x - y}{v} }[/tex] (orange); w = 3

I. x = [tex]\frac{y - y_1 }{m} + x_1[/tex] (pink); x = 1

J. c = 5a - 7 (brown); c = 9

To solve for a variable in an equation, we need to isolate the variable to one side of the equation.

A. a = b/c²

a(c²) = b/c²(c²) [multiplication property of equality]

ac² = b

b = ac² (dark green)

Substitute a = 1/8 and c = 4 into the equation

b = (1/8)(4²)

b = (1/8)(16)

b = 2

B. m - v = k

m - v - m = k - m [subtraction property]

-v = k - m

v = -k + m

v = m - k (light blue)

Substitute k = -7 and m = -2 into the equation

v = -2 - (-7)

v = -2 + 7

v = 5

C. p/q = r/s

Cross multiply

qr = ps

qr/r = ps/r [division property]

q = ps/r (yellow)

Substitute k = -7 and m = -2 into the equation

q = (14)(-4)/-7

q = 8

D. 4m + 2n = 5n

4m + 2n -2n = 5n - 2n [subtraction property]

4m = 3n

4m/4 = 3n/4 [division property]

m = 3n/4 (red)

Substitute n = 8 into the equation

m = 3(8)/4

m = 6

E. w = x/y - z

w + z = x/y - z + z [addition property]

w + z = x/y

y(w + z) = x/y(y) [multiplication property]

y(w + z) = x

x = y(w + z) (light green)

Substitute w = -8, y = -5, and z = 6 into the equation

x = -5(-8 + 6)

x = -5(-2)

x = 10

F. m = 1/2(d1 + d2)

2m = d1 + d2 [multiplication property]

2m - d2 = d1 [subtraction property]

d1 = 2m - d2 (dark blue)

Substitute m = 10 and d2 = 13 into the equation

d1 = 2(10) - 13

d1 = 20 - 13

d1 = 7

G. 2r = 5s/2 - s/4

2r = (10s - s)/4

2r = 9s/4

2r(4) = 9s/4(4) [multiplication property]

8r = 9s

8r/9 = 9s/9 [division property]

8r/9 = s

s = 8r/9 (purple)

Substitute r = 9/2 into the equation

s = 8/9(9/2)

s = 4

H. vw² + y = x

vw² + y - y = x - y

vw² = x - y

vw²/v = (x - y)/v

w² = (x - y)/v

w = [tex]\sqrt{\frac{x - y}{v} }[/tex] (orange)

Substitute v = 5, x = 38, y = -7 into the equation

w = [tex]\sqrt{\frac{38 - (-7)}{5} }[/tex]

w = √9

w = 3

I. [tex]y - y_1 = m(x - x_1)[/tex]

[tex]\frac{y - y_1 }{m} = x - x_1\\\\\frac{y - y_1 }{m} + x_1 = x[/tex]

x = [tex]\frac{y - y_1 }{m} + x_1[/tex] (pink)

Substitute x1 = 9, y = 19, y1 = -5, and m = -3 into the equation

x = (19 - (-5))/-3 + 9

x = -8 + 9

x = 1

J. 2a + 3c = 17a - 21

2a - 17a = -3c - 21

-15a = -3c - 21

-15a + 21 = -3c

-15a/-3 + 21/-3 = -3c/-3

5a - 7 = c

c = 5a - 7 (brown)

Substitute a = 16/5 into the equation

c = 5(16/5) - 7

c = 16 - 7

c = 9

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The required value of sports imports of the United States in 2012 is $2,745,240,000,000.

.

An expression is a number, or a variable, or a combination of numbers and variables and operation symbols.

Now it is given that,

the total value of exports from the United States in 2012 = $2,210,585,000,000

Now it is also given that,

the total value of exports from the United States in 2012 = $837,965,000,000 more than half of the US imports

Therefore, we can write

the total value of exports from the United States in 2012 = $837,965,000,000 + imports/2

Putting the value of exports we get,

2,210,585,000,000 = 837,965,000,000 + imports/2

Taking alike terms together we get,

Imports/ 2 = 2,210,585,000,000 - 837,965,000,000

Solving we get,

Imports/ 2 = 1,372,620,000,000

Multiplying by 2 both the side by 2 we get,

Imports = 2,745,240,000,000

Thus, the value of imports of the United States in 2012 = $2,745,240,000,000

Thus, the required value of imports of the United States in 2012 is $2,745,240,000,000.

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The correct question is :

in 2012 the total value of exports from the United States was $2,210,585,000,000 that year exports were $837,965,000,000 more than half of the US imports. find the value of imports of the United States In 2012

The string can be cut into 50 pieces of 10 cm length.

Length is a measurement of how long an object is or the distance between two points. Length is the term used to identify the size of an object or the distance from one point to another. It can be used to determine an object's size or the distance between two points.

A measurement of length is the distance between two locations or the length of an item. The term "length" is used to describe an object's size or the separation between two points. It may be used to calculate the size of an item or the separation between two places. Distance is measured in length. Length has the dimension of distance in the International System of Quantities. The majority of measuring systems choose a base unit for length from which all other units are derived.

Centimeters, meters, kilometers, inches, feet, and yards are some examples of conventional units used to measure length.

1 m = 100 cm

so, no. of pieces = 500/10 = 50.

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Answer: 50 pieces

Step-by-step explanation:

      First, we need both of the measurements to be in the same unit.

5 m ➜ 500 cm

      Next, we will use division to see how many 10 cm pieces of string can be cut.

500 cm / 10 cm = 50 pieces

The diameter of the given sphere is; 39.5 cm

The formula for Volume of Sphere is;

V = 4/3 πr³

where;

V is volume

r is radius

We are given; V = 32250 cm³

Thus;

4/3 πr³ = 32250

Make r the subject of the formula to get;

r = ∛((32250 * 3/4)/π)

r =  ∛7699.12

r = 19.75 cm

Now, radius * 2 = diameter

Thus; diameter = 2 * 19.75 = 39.5 cm

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Given the data in the table, the slope, y-intercept and equation of the table are 1/3, (0,-2), and y = (1/3)x - 2 respectively.

Given that;

x           y

-3        -3

0         -2

3         -1

6         0

9         1

Slope m = rise/run = (y₂ - y₁)/(x₂-x₁)

Using the, the points (-3,-3) and (0,-2) from the table

Slope m = rise/run = (y₂ - y₁)/(x₂-x₁)

Slope m = rise/run = ((-2) - (-3)) / ( 0 - (-3) )

Slope m = 1/3

To determine the y-intercept b, use the formula for equation of line.

y = mx + b

Substitute in Point( -3,-3 ) and m = 1/3 and solve for b

y = mx + b

-3 = (1/3)(-3) + b

-3 = -3/3 + b

-3 = -1 + b

b = -3 + 1

b = -2

Hence the y-intercept in point form is (0,-2)

The equation of the line in slope-intercept form will be;

y = mx + b

y = (1/3)x + (-2)

y = (1/3)x - 2

Given the data in the table, the slope, y-intercept and equation of the table are 1/3, (0,-2), and y = (1/3)x - 2 respectively.

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Answer:

He has to pay $223.68 after 6 months

Step-by-step explanation:

I don't know if I understood you right but I think you meant how much does he have to pay in 6 months. That would be 37.28x6=223.68

Answer:

Step-by-step explanation:

30.466666.999

2.81*10^4

the number of times you move the decimal adds to the exponent

The value of the ratio is CD = 24 cm

The given parameters are

SR : CD = 3 : 4

SR = 18 cm

Substitute SR = 18 cm in the ratio SR : CD = 3 : 4

So, we have

18 cm : CD = 3 : 4

Express the ratio as a fraction

So, we have

18 cm/CD = 3/4

This gives

CD = 18 cm * 4/3

Evaluate

CD = 24 cm

Hence, the value of the ratio is CD = 24 cm

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The value given expression of (x⁵ + 8)/(x + 1) is (x⁶ + 8x).

The given expression is-

(x⁵ + 8)/(x + 1)  .......equation 1

It is stated that;

x² + x = 1

Taking x common;

x(x + 1) = 1

Bring x on the other side.

(x + 1) = 1/x

Put the value of (x  + 1) in equation 1.

= (x⁵ + 8)/(x + 1)

= (x⁵ + 8)/[1/x]

Simplifying;

= (x⁵ + 8)x

Multiplying x on both values;

= x⁵x + 8x

= x⁶ + 8x

Thus, the value of the given expression is (x⁶ + 8x).

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The number is 90.

Percentage can be described as the fraction of an amount that is expressed as a number out of hundred. The sign used to represent percentage is %. Percentages are a measure of frequency.

When the number is decreased, the number becomes smaller. This means that the original number is larger than 81.

The equation that can be used to represent the information in the question is:

(1 - percentage decrease) initial number = 81

(1 - 0.1)x = 81

0.9x = 81

x = 81 / 0.9

x = 90

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ln (e) = 1, by the properties of logarithm.

Now,

As seen by the definition of logarithms, [tex]log_b(a)=x[/tex]

In case of natural logarithm, b = e, i.e., [tex]log_e(a)=x[/tex] = ln (a) = x

In the given expression, a = e

Hence, ln (e) = x

By properties of logarithms, we get that: ln (e) = 1.

Hence, ln (e) = 1, by the properties of logarithm.

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Answer:

Step-by-step explanation:cevap çok basit cevap eşittir cevap yani çok basit

Answer:

3^x=x^(x/3)

3^(3*x)=x^(3*x/3)

3^(3x)=x^x

(3^3)^x=x^x

27^x=x^x

27^(x*1/x)=x^(x*1/x)

27^(x/x)=x^(x/x)

27^1=x^1

x=27

Step-by-step explanation:

Step-by-step explanation:

[tex] = - 8( - 6) - (25) \\ = 48 - 25 \\ = 23[/tex]

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The three undefined terms that form the basis for geometry are:

- Point

- Line

- Plane

The term undefined is often used to refer to an expression that is not assigned an interpretation or a value.

The main three undefined terms of geometry are:

- Point

In geometry, a point has no dimension.

A point is represented by a dot.

The point has no length, width, or thickness.

A dot can be very tiny or very large.

- Line

A line is a straight one-dimensional figure that has a length but does not have a thickness, and it extends endlessly in both directions.

A line is made of a set of points that is extended in opposite directions infinitely.

- Plane

A plane, in geometry, prolongs infinitely in two dimensions.

It has zero thickness, zero curvature, infinite width, and infinite length.

It is also known as a two-dimensional surface.

Thus,

The three undefined terms that form the basis for geometry are:

- Point

- Line

- Plane

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