The binomial expansion for ( X + Y )⁴ is X⁴ + 4 X³(Y) + 6 X²(Y²) + 4 XY³ + Y⁴.
Use the binomial expansion theorem to find each term. The binomial theorem states ( a + b )ⁿ = ⁿ∑ _{k = 0} n C k ⋅ ( a^(n - k) × b^{k} ) .
⁴∑ _{k = 0} = ( 4 ! / [ ( 4 − k ) ! k ! ] ) × ( X )^{4 - k} × ( Y )^{k}
Expanding the summation.
( X + Y )⁴ = ( 4! / ( 4 − 0 )! 0! ) × ( X )⁴⁻⁰ × ( Y )⁰ + ( 4! /( 4 − 1 )! 1! ) × ( X )⁴⁻¹ × ( Y )¹ + ( 4! / ( 4 − 2 )! 2! ) × ( X )⁴ ⁻ ² × ( Y )² + ( 4! /( 4 − 3 )! 3! ) × ( X )⁴ ⁻ ³ × ( Y )³ + 4! / ( 4 − 4 )! 4! × ( X )⁴ ⁻ ⁴ × ( Y )⁴
Simplify the exponents for each term of the expansion.
( X + Y )⁴ = 1 × ( X )⁴ × ( Y )⁰ + 4 × ( X )³ × ( Y )¹ + 6 × ( X )² × ( Y )² + 4 × ( X )¹ × ( Y )³ + 1 × ( X )⁰ × ( Y )⁴
Therefore,
( X + Y )⁴ = X⁴ + 4 X³ (Y) + 6 X² (Y²) + 4 X Y³ + Y⁴
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cuanto es 0,006720 dividido entre 0,007500
Answer:
0,896
Step-by-step explanation:
bought a used car for $7,500. He had $4600 and borrowed the rest from his parents. If it took
Johnny 8 months to pay back the money he borrowed, what was the monthly payment assuming that each month
he paid the same amount?
Answer: The monthly payment assuming that each month he paid the same amount is = 362.5 $
Step-by-step explanation:
Given data,
Johnny bought a used car for $7,500
He had $4600 and borrowed the rest from his parents.
Johnny 8 months to pay back the money he borrowed
So, we can write ,
Let us assume, x1 = $7,500 and x2 = $4600
Based on the given conditions,
we use formula is :
x = ( x1 - x2 ) / pay back months
x = ( 7,500 - 4,600 ) / 8
we can subtract the values,
we can write,
x = 2900/8
x = 725/2
x = 362.5 $
Therefore,
The monthly payment assuming that each month he paid the same amount is = 362.5 $.
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ppppppllllllssss hhhhheeeeellllppppp
The exponent -6⁻⁴ is -(1/6⁴) which simplifies to -(1/1296).
What is exponent?Exponents represent how many times a number is multiplied by itself. Numbers written in exponent form are more compact for a large number of repetitive multiplications.
We know moving an exponent from numerator to denominator or vice versa changes the sign of the exponent.
∴ -6⁻⁴ can be written as -(1/6⁴) which is,
-(1/1296).
Now, simplifying (-6)⁻⁴ which is 1/(-6)⁴,
= (1/1296).
So, their numerical value is the same but they are of different signs as (-6) multiplied 4 times gives a positive number.
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lf f(x) = 4x³ + Ax² + 7x − 1 and f(2) = 7, what is the value of A?
Answer:
4(2)³+A×2²+7(2)-1=7
4×8+4A+14-1=7.
32+14-1-7=-4A..
38=-4A.
A=38/-4.
A=-9½
Solve each proportion. 10/14=15/x
Answer:x=21
Step-by-step explanation:
10/14=15/x
5/7=15x^-1
(5/7)/15=ans
ans=x^-1
ans^-1=21
Answer: x=21
Step-by-step explanation:
[tex]\frac{10}{14} =\frac{15}{x}[/tex]
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 14x, the least common multiple of 14,x.
x×10=14×15
Multiply 14 and 15 to get 210.
x×10=210
Divide both sides by 10.
x=
10
210
Divide 210 by 10 to get 21.
x=21
The value of an industrial machine has a decay factor of 0.75 per year. After six years, the machine is worth $ 7500 . What was the original value of the machine?
The original value of the machine is $ 30 , 720 , 000 .
Given : Decay factor = 0.75 per year
Present value of machine ( after 6 yrs . ) = $ 7500 .
According to ques .
The original value of machine be worth $ x .
After nth yr. the present value of machine will be
( x - 0.75 x ) [ [tex]( 1 - 0.75 ) ^{n-1}[/tex] ] .
Therefore , after 6 yrs. the present value of machine will be
( x - 0.75 x ) [ [tex]( 1 - 0.75 ) ^{5}[/tex] ] = 7500
0.25 x * ( [tex]0.25 ^{5}[/tex] ) =7500
Hence , x = 30 , 720,000
Therefore , the original value of the machine is $ 30 , 720 , 000 .
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Joy recently hired a landscaper to do some necessary work. On the final bill, Joy was charged a total of
$
1215
.
$
445
was listed for parts and the rest for labor. If the hourly rate for labor was $70, how many hours of labor was needed to complete the job?
Joy was charged the labor of 11 hours
Joy was charged a total of $1215 from the landscaper for listed parts and labor
He was charged $445 for listed parts
He was charged the rest for labor
The hourly rate for labor is $ 70
Let the number of hours worked by labor be x
so, we can form a linear equation here as 445 + 70x = 1215
Now we have to solve this linear equation
445 + 70x = 1215
70x= 1215 - 445
70x = 770
x = 770/ 70
x = 11
Therefore, 11 hours of labor was needed to complete the job.
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The sum of thirty six and fourteen and divided by the sum of one and three
pppppllllllsssss hhhhheeelllppppp i need this rn
The perimeter of an equilateral triangle is 9 inches more then the perimeter of a square, and side of the triangle is 6 inches longer then the side of the square. Find the side triangle.
Answer:
The sides of the triangle are 15 inches and the sides of the square are 15 - 6 = 9 inches
Step-by-step explanation:
It is given that the perimeter of an equilateral triangle is 9 inches more then the perimeter of a square, and side of the triangle is 6 inches longer then the side of the square.
Now,
An equilateral triangle has all 3 sides the same length
Let one side of the triangle be x so the perimeter will be 3x.
Further, if one side of the square is 6 longer than a side of the triangle
then the square has side length x - 6 and perimeter is;
=> 4(x - 6)
=> 4x - 24
Since, the perimeter of the triangle is 9 more than the perimeter of the square we get the equation as;
=> 3x = 4x - 24 + 9
=> x = 15
Therefore,
The sides of the triangle are 15 inches
The sides of the square are 15 - 6 = 9 inches
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11. The U.S. Census Bureau has a population cloc
on the Internet. On a recent day, the United
States population was listed as 310,763,136.
Write this number in word form
The mentioned number in word form is three hundred ten, seven hundred sixty three, one hundred thirty six.
What is a number?The arithmetic value of a number is one that is used to denote quantity. As a result, a number is a mathematical concept used for counting, measuring, and labeling. As a result, mathematics is based on numbers.
Early humans used a variety of symbols to represent numbers, as evidenced by the inscriptions discovered at archaeological sites. For instance, farmers, traders, and merchants in antiquity used tally marks to indicate quantities. A standing line is drawn for each count in tally marks, and the fifth count is indicated by crossing out the first four lines. However, this was a laborious method and it was impractical to display quantities.
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Aubrey is flying a kite, holding her hands a distance of 2.5 feet above the ground and
etting all the kite's string play out. She measures the angle of elevation from her
and to the kite to be 25°. If the string from the kite to her hand is 150 feet long, how
many feet is the kite above the ground? Round your answer to the nearest hundredth
of a foot if necessary.
The distance of kite above the ground is 65.33 feet.
What are the six trigonometric ratios?Trigonometric ratios for a right angled triangle are from the perspective of a particular non-right angle.
In a right angled triangle, two such angles are there which are not right angled(not of 90 degrees).
The slant side is called hypotenuse.
From the considered angle, the side opposite to it is called perpendicular, and the remaining side will be called base.
From that angle (suppose its measure is θ),
[tex]\sin(\theta) = \dfrac{\text{Length of perpendicular}}{\text{Length of Hypotenuse}}\\\cos(\theta) = \dfrac{\text{Length of Base }}{\text{Length of Hypotenuse}}\\\\\tan(\theta) = \dfrac{\text{Length of perpendicular}}{\text{Length of base}}\\\\\cot(\theta) = \dfrac{\text{Length of base}}{\text{Length of perpendicular}}\\\\\sec(\theta) = \dfrac{\text{Length of Hypotenuse}}{\text{Length of base}}\\\\\csc(\theta) = \dfrac{\text{Length of Hypotenuse}}{\text{Length of perpendicular}}\\[/tex]
We are given that;
Distance = 2.5 feet
Now,
Using this function, you can write an equation for the height of the kite, h:
sin(25°) = h / 150
Solving for h, you get:
h = 150 × sin(25°)
h = 62.83 feet
However, this is not the final answer, because you also need to add Aubrey’s hand height above the ground, which is 2.5 feet. Therefore, the total height of the kite above the ground is:
h + 2.5 = 62.83 + 2.5
h + 2.5 = 65.33 feet
Rounding to the nearest hundredth of a foot, you get:
h + 2.5 = 65.33 feet
Therefore, by trigonometric ratios the answer will be 65.33 feet .
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Answer:
The kite is 65.89 feet above the ground.
Step-by-step explanation:
Write an equation in slope-intercept form of the line that passes through (-9, -3) and (-7, 5)
The slope-intercept form of the line passing through the two points is
y = 4x + 33
Slope-intercept equation is one of the special forms of the linear equation.
To find the slope,
The point-slope formula is used to find the slope
y₂-y₁ = m (x₂-x₁)
where (x₁,y₁) and (x₂,y₂) are (-9,-3) and (-7,5), Then
5 - (-3) = m [-7 - (-9)]
5 + 3 = m (-7+9)
8 = m(2)
m = 8/2
m = 4
Therefore, the slope of the line passing through two points is 4
To find the intercept,
We can use the slope intercept formula to find the intercept
y = mx + b
where (x,y) be (-9,-3) abd m = 4 , Then
-3 = 4(-9) + b
-3 = -36 = +b
b = 36-3
b = 33
Therefore, the intercept of the line passing through two points is 33
Thus, the slope intercept form of the line passing through the two points
(-9, -3) and (-7, 5) can be written as
y = 4x + 33
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Sequences: sort and solve
recording document
scenario
a. mallory's new year resolution is to become a better runner. due to the frigid
michigan temperatures and her extreme commitment to her studies, mallory is
unable (or unmotivated) to start working on her goal until school gets out. on
june 1st, she starts by running 15 minutes and every day after that she adds 5
minutes to her run.
write an explicit formula and a recursive formula
An explicit formula for Mallory's run will be 15 + 5x.
What is an expression?
Mathematical expression is the combination of numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also be used to denote the logical syntax's operation order and other properties.
Given that on June 1st, she starts by running for 15 minutes, and every day after that she adds 5 minutes to her run.
Equation = 15 + 5x, where x is the number of days.
Therefore, the explicit formula for Mallory's run will be 15 + 5x.
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Meghan needs to create a 4-digit password. The digits cannot repeat in the password and cannot start with the digit 0. What is the probability , to the nearest hundredth of a percent, the the password starts with a 9 and ends with a 6?
The probability to the nearest hundredth of a percent, the the password starts with a 9 and ends with a 6 is 1%.
A four digit password will have number ranging from 0-9.
There are four spots in total.
The first spot can not be taken by zero, so only 9 options can take the place of first spot. As the number can not be repeated, the second spot has only options a 0-8. The third spot will eventually have 8 options only. The Fourth spot will have 7 options available to fill the spot.
Hence total number of 4-digit passwords combinations can be calculated by multiplying the options,
= 9 x 9 x 8 x 7
= 10206.
A total of 10206 passwords are possible.
The number of passwords starting from nine and ending with six can be found as,
= 1 x 9 x 8 x 1
= 72
A total of 72 passwords starting from 9 and ending with 6.
The probability that a password chosen is starting from 9 and ending with 6 is,
P(E) = 72/10206
P(E) = 0.007
Probability in percent,
= 0.007 x 100
= 0.7
Rounding it off, we get nearly 1%.
So, the probability is nearly 1%.
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Write each expression as a single natural logarithm. 4 ln 3
Expression as a single natural logarithm of 4 ln 3 is 4.39444915467
What is natural logarithm?
4 ln 3
=[tex]ln 3^{4}[/tex]
= ln 81
=4.39444915467
The inverse of an exponential function, the natural log is the logarithm to the base of the number e. Natural logarithms are unique varieties of logarithms that are employed in the treatment of time and growth-related issues.
The distinction between log and ln is that log is expressed in terms of base 10, while ln is expressed in terms of base e. As an illustration, log of base 2 is denoted by log2 and log of base e by loge = ln (natural log).
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ALGEBRA I A/2. Foundational Concepts / 2.4. Properties of Real Numbers
X
Which best describes your ability to use a number line to graph, order, and compare real numbers?
OA. I can use a number line to graph, order, and compare real numbers. I can teach someone else how to do this as well.
OB. I can use a number line to graph, order, and compare real numbers. Once in a while I make a mistake.
OC. I can do some of these things, but not all of them. I make some mistakes.
OD. I am not able to work with number lines. I need help.
A number line plot is a graph that displays the frequency a number occurs in a set of data. Dot plot is another name for a number line plot.
The number line is a one-dimensional graph that can display all of the rational numbers (whole numbers, integers, and rational numbers).
The numbers we use for counting, or enumerating items, are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe them in set notation as {1,2,3,...} where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero , 0,1,2,3,...
The set of integers adds the opposites of the natural numbers to the set of whole numbers: −3,−2,−1,0,1,2,3,
It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.,−3,−2,−1,negative integers
0,zero1,2,3, positive integers.
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Solve each equation. Check for extraneous solutions. √x+3 -1=x
the extraneous solution for the equation √ (x + 3) - 1 = x is x = 1 or x = - 2.
We are given an equation:
√ (x + 3) - 1 = x
√ (x + 3) = x + 1
Raise both sides to the power 2, we get that:
x + 3 = x² + 1 + 2 x
x² + 2 x - x + 1 - 3 = 0
x² + x - 2 = 0
x² + 2 x - x - 2 = 0
x ( x + 2) - 1 ( x + 2) = 0
(x - 1)(x + 2) = 0
x = 1 or x = - 2.
Therefore, we get that, the extraneous solution for the equation √ (x + 3) - 1 = x is x = 1 or x = - 2.
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Find the degree measure of an angle of 3 π/4 radians.
Answer: 135°
Step-by-step explanation:
Since π = 180°
(3 × π)/ 4
Substitute 180° for π
= (3 × 180)/4
Simplify
= 540/4
= 135°
Solve for ttt:-\dfrac{7}{4}=\dfrac{2}{5}t− 47 = 52 t
Answer: Solution is t = 905/8
Step-by-step explanation:
Given data,
ttt:-\dfrac{7}{4}=\dfrac{2}{5}t− 47 = 52 t
We can write,
- 7/4 = (2/5) t - 47
Rearrange variables to the left side of the equation
- (2/5) t = - 47 + 7/4
Find common denominator and write the numerators above common denominator
- (2/5) t = ( -188 + 7 ) / 4
calculate the sum or difference
- (2/5) t = -181/4
Divide both sides of the equation by the coefficient of variable
t = (-181/4) x (-5/2)
Determining the sign for multiplication or division
t = 181/4 x 5/2
Write as a single fraction
t = ( 181 x 5 ) / ( 4 x 2 )
Calculate the product or quotient
t = (905/8)
Therefore, t = 905/8
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Patrick grated 2/3 of a block of cheese to make macaroni. What fraction of the block is left
100 points top anyone who can answer these questions.
in the given figure the value of x is 45°.
What is angle?In Plane Geometry, a figure which is formed by two rays or lines that shares a common endpoint is called an angle and corner whether constituting a projecting part or a partially enclosed space.
Given:
<PQR= x+7
<SQR=x+3
<PQR+<SQR=<PQS=100°
According to given question we have
<PQR+<SQR=<PQS
Put the value we get
x+7+x+3=100
Compare like terms we get
2x+10=100
Substract 10 from both sides we get,
2x=90
Divide 2 on both sides we get,
x=45°
Therefore, in the given figure the value of x is 45°.
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Brianna asked 45 students if they would vote for her to be student council president. She used her calculator to compare
the number of students who said yes with the total number of students. Her calculator showed the result as 0.6222....
Answer parts a and b.
Write this number as a fraction.
Brianna would receive 28 votes from students.
What are fractions?Any number of equal parts is represented by a fraction, which also represents a portion of a whole. A fraction, such as one-half, eight-fifths, or three-quarters, indicates how many parts of a particular size there are when spoken in everyday English. There are three main categories of fractions in mathematics. Proper fractions, incorrect fractions, and mixed fractions are these three types. The terms with a numerator and a denominator are called fractions.So, a recurring decimal at the number "2" is "0.6222.
Let x = 0.62222....
Multiplying by 10 to keep the repeating digits after the decimal point, x by 10x equals 6.2222.Multiplying by 100 allows us to once more leave the recurring digit after the decimal point, so multiplying x by 100 results in 62.2222.Subtract 10x from 100x as follows:
100x - 10x = 62.2222... - 6.2222... 90x = 56x = 56/90x = 28/45Therefore, Brianna would receive 28 votes from students.
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The correct question is given below:
Brianna asked 45 students if they would vote for her to be student council president. She used her calculator to compare the number of students who said yes with the total number of students. Her calculator showed the result as 0.6222...
1. Write this number as a fraction (Steps)
2. How many students said they would vote for Brianna?
Find the y-intercept and x-intercept of the line.
-x+2y=2
Written in simplest exponential form the product (3x¹)(-2x¹)
(1) –5r2
(3) x¹
(2) -6x¹0
(4)-6x²1
is?
I need a little help...
Answer:
10 pencils
Step-by-step explanation:
16-x = 36-3x
16-36 = -3x+
thus x = 10
What is the decimal equivalent of 2 2/4
Answer:
2.5
Step-by-step explanation:
2 2/4
Simplify 2/4
2/4 = 1/2 = 0.5
2 2/4 = 2 + 0.5 = 2.5
2 2/4 = 2 1/2
2 1/2 = 2.50 (or just) 2.5
Write each expression as a single logarithm.
log₂9-log₂3
If the expression is log₂9-log₂3, then it will look like [tex]log_{2}3[/tex] in single logarithm.
Given that the expression is log₂9-log₂3.
We have to give the single logarithm for the given expression.
The quotient rule for logarithms tells us that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule we also can use the inverse property to derive the quotient rule. The quotient property basically says that log (m/n)=log m-log n.
The expression is log₂9-log₂3.
log₂9-log₂3=[tex]log_{2}[/tex](9/3)
We can write it as [tex]log_{2}3[/tex].
Hence if the expression is log₂9-log₂3, then it will look like [tex]log_{2}3[/tex] in single logarithm.
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1. A line segment has endpoints of Y(2,-3) and Z(-6,-1).
a. Calculate the length of the line segment.
b. Calculate the midpoint of the line segment.
c. Determine the equation of the line that passes through the points Y and Z.
The answers will be :
a. The length of the line segment will be equal to 8.24 units.
b. The midpoint of the line segment will be ( -2 , -2 ).
c. The equation of line passing through points Y and Z will be 4y + x - 10 = 0
What is slope of a line segment ?
The ratio of the difference in y-coordinates over the equivalent x-coordinates between two different locations on a line.
It is given that a line segment has endpoints of Y(2,-3) and Z(-6,-1).
Let's solve the given parts based on the above data.
a.
The length of the line segment will be given by :
YZ = [tex]\sqrt({x_{2} - x_{1})^2 + (y_{2}- y_{1})^2}[/tex]
YZ = √ [( -6 -(2)]^2 + [ -1 - (-3)]^2}
YZ = √ (-8)² + 2²)
YZ = √68
YZ = 8.24 units
b.
The midpoint (say M) of the line segment will be given by :
M = ( [tex]\frac{x_{1} + x_{2} }{2} , \frac{y_{1} + y_{2} }{2}[/tex])
M = [(2-6)/2 , (-3-1)/2]
M = ( -2 , -2 )
c.
And the equation of line passing through points Y and Z will be :
[tex]y_{2} - y_{1}[/tex] = m ( [tex]x_{2} - x_{1}[/tex] )
Let's calculate value of slope (m) firstly which will be :
m =([tex]y_{2} - y_{1}[/tex]) / ( [tex]x_{2} - x_{1}[/tex] )
m = ( -1 + 3 ) / (-6 - 2)
m = 2 / -8
m = -1 / 4
or
m = -0.25
Using the value of slope (m) : we get the equation of line as :
[tex]y - y_{1}[/tex] = - 0.25 ( [tex]x - x_{1}[/tex] )
or
(y + 3) = - 0.25 ( x - 2)
y + 3 = -0.25 x + 0.50
or
y + 0.25 x - 2.5 = 0
If we multiply by 4 throughout the equation ; then the equation of line can also be written as :
4y + x - 10 = 0
Therefore the answers will be :
a. The length of the line segment will be equal to 8.24 units.
b. The midpoint of the line segment will be ( -2 , -2 ).
c. The equation of line passing through points Y and Z will be 4y + x - 10 = 0
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Four-thirds of a number is decreased by 12. The result is eight
Answer:
The number is 15
Step-by-step explanation:
x = a number
4/3 x - 12 = 8
Rearrange the equation so that the integers are on one side - in the format Ax = B
4/3 x - 12 = 8
4/3 x = 8 + 12
4/3 x = 20
Simplify the equation - in the format x = A - then solve for x
4/3 x = 20
4 x = 20 * 3
4 x = 60
x = 60 / 4
x = 15
[tex] \large \bf \implies{A\: number\: is\:15}[/tex]
Step-by-step explanation:A number is x
then we have [tex]\frac{4}{3}[/tex]x - 12 = 8
[tex] \bf \longrightarrow \frac{4}{3}x - 12 = 8 \\ \\ \bf \longrightarrow \frac{4}{3}x = 8 + 12 \\ \\ \bf \longrightarrow \frac{4}{3}x = 20 \\ \\ \bf \longrightarrow 4x = 20 \times 3 \\ \\ \bf \longrightarrow4x = 60 \\ \\ \bf \longrightarrow x = \frac{ \cancel{60}}{ \cancel{4} } \\ \\ \bf \longrightarrow \: x = 15[/tex]
'Four thirds of the number is [tex]\frac{4}{3}x[/tex] then decreased by 12 is [tex]\frac{4}{3}x - 12[/tex] then gives the result 8. That is [tex]\bf\frac{4}{3}x - 12 = 8[/tex]'