Answer:
32a^5+160a^4b+320a^3b^2+320a^2b^3+160ab^4+32b^5
============================================================
Explanation:
Let's use Pascal's Triangle
In the row that starts with 1,5,... we have the values 1,5,10,10,5,1
These will be the coefficients of the terms.
Let x = 2a and y = 2b
We want to expand out (x+y)^5
Using pascals triangle, we get the following expansion
(x+y)^5 = 1x^5y^0+5x^4y^1+10x^3y^2+10x^2y^3+5x^1y^4+1x^0y^5
The numbers in bold are the coefficients 1,5,10,10,5,1 found earlier.
Note how the exponents for x start at 5 and count down to 0; while the y exponents start at 0 and count up to 5. For any term, the x and y exponents always add to 5 for the expansion of (x+y)^5. In general, the exponents of any term will add to n for (x+y)^n.
At this point, we plug in x = 2a and y = 2b
Since this will clutter things a bit, I'll do it term by term
1x^5y^0 = 1(2a)^5(2b)^0 = 1(32a^5)(1) = 32a^55x^4y^1 = 5(2a)^4(2b)^1 = 5(16a^4)(2b) = 160a^4b10x^3y^2 = 10(2a)^3(2b)^2 = 10(8a^3)(4b^2) = 320a^3b^210x^2y^3 = 10(2a)^2(2b)^3 = 10(4a^2)(8b^3) = 320a^2b^35x^1y^4 = 5(2a)^1(2b)^4 = 5(2a)(16b^4) = 160ab^41x^0y^5 = 1(2a)^0(2b)^5 = 1(1)(32b^5) = 32b^5So in the end, the expression (2a+2b)^5 expands out to
32a^5+160a^4b+320a^3b^2+320a^2b^3+160ab^4+32b^5
The binomial theorem uses the same basic idea, but instead of using Pascal's Triangle to get the coefficients, you'll use the nCr combination formula.
Answer: [tex](2a+2b)^5 =32a^5+160a^4b+320a^3b^2+320a^2b^3+160ab^4+32b^5[/tex].
Using Pascal's triangle we get:
[tex](x+y)^5 = 1x^5y^0+5x^4y^1+10x^3y^2+10x^2y^3+5x^1y^4+1x^0y^5[/tex]
Here, x = 2a and y = 2b
So the terms become:
[tex]1x^5y^0 = 1(2a)^5(2b)^0 = 1(32a^5)(1) = 32a^5\\5x^4y^1 = 5(2a)^4(2b)^1 = 5(16a^4)(2b) = 160a^4b\\10x^3y^2 = 10(2a)^3(2b)^2 = 10(8a^3)(4b^2) = 320a^3b^2\\10x^2y^3 = 10(2a)^2(2b)^3 = 10(4a^2)(8b^3) = 320a^2b^3\\5x^1y^4 = 5(2a)^1(2b)^4 = 5(2a)(16b^4) = 160ab^4\\1x^0y^5 = 1(2a)^0(2b)^5 = 1(1)(32b^5) = 32b^5\\[/tex]
Hence: [tex](2a+2b)^5 =32a^5+160a^4b+320a^3b^2+320a^2b^3+160ab^4+32b^5[/tex]
Learn more: https://brainly.com/question/13769924
If x=5 and the area of the rectangle is 30 what is the other length of the rectangle?
Answer:
The other length would be 6.
Step-by-step explanation:
Rectangle area is l times height. So x is either the length for height. So we have 5 times x = 30. To find the other length we have to take 30/5 which equals 6
If f(x)=−3x2−2x+1, find 2[f(r)]
Please use your BIG BRAIN SKILLS this is the last question on my test please help..... :{
Answer:
[tex]-6x^2r-4xr+2r[/tex]
Step-by-step explanation:
[tex]f(x)=-3x^2-2x+1\\\\2[f(r)][/tex]
Plug in the value of f of x into the second expression:
[tex]2[(-3x^2-2x+1)(r)][/tex]
Use the distributive property to distribute the two:
[tex]2(-3x^2)+2(-2x)+2(1)[r]\\\\-6x^2-4x+2[r][/tex]
Distribute each value among r:
[tex]-6x^2r-4xr+2r[/tex]
:Done
i want to purchase a necklace for 325$ . ive already saved 115.00 and can set aside 22.00$ a week . which inequality can be used to find the number of weeks it will take to save at least 325.00$?
Answer:
22w + 115 = 325 w = about 10 weeks
Step-by-step explanation:
The first part of this inequaltiy is 22w. Since he/she will set aside $22 a week, you would put a variable because the number of weeks you will need to set aside $22 is unknown therefore you put a variable that makes sense. I put w because you can say that w=weeks. The next part of the inequality is adding $115 because he/she has already saved $115 so we add that to 22w. Finally 22w+115 has to equal $325 because that is how much the necklace costs. Hope this helps!
f(4) if f(x) = 3x - 1
Simply substitute the value of 3 given. In other words,
f(3) is the function of
f(x), however, in this case, x is 3
f(3)=3(3)−4
Distribute 3 through the parenthesis
f(3)=9−4
Simplify
f(3)=5
please help, we have 0 brain cells left
Answer:
no solution
Step-by-step explanation:
Question in picture solve
Answer:
x= 13.5
Step-by-step explanation:
f(x) = 2x + 9
f(x) = 36
Set the 2 equations equal to each other;
2x + 9 = 36
Subtract 9 from both sides;
2x = 27
Divide both sides by 2;
x = 13.5
Check:
f(x) = 2x + 9
f(13.5) = 2(13.5) + 9= 27 + 9 = 36
Write the equation of the line fully simplified slope intercept form
Answer:
y = - 5x - 5
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Calculate m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (- 1, 0) and (x₂, y₂ ) = (0, - 5) ← 2 points on the line
m = [tex]\frac{-5-0}{0+1}[/tex] = - 5
The line crosses the t- axis at (0, - 5) ⇒ c = - 5
y = - 5x - 5 ← equation of line
A generator can run for 11 hours on 3 gallons of gasoline. How many hours can the generator run on 9 gallons of gasoline?
Answer:
11 / 3 hours
Step-by-step explanation:
Write a slope-intercept form equation for a line that passes through (0,4) and (2,-2).
Show the step by step process please.
Answer:
-3
Step-by-step explanation:
Slope is essentially change in height over change in horizontal distance, and is often referred to as "rise over run." It has applications in gradients in geography as well as civil engineering, such as the building of roads. In the case of a road the "rise" is the change in altitude, while the "run" is the difference in distance between two fixed points, as long as the distance for the measurement is not large enough that the earth's curvature should be considered as a factor. The slope is represented mathematically as:
m =
y2 - y1
x2 - x1
someone help please will give brainliest
5x - 7 = 8x + 14
I will give 30 points
please do not just answer for points i actually really need help fast
Answer: -7
Step-by-step explanation:
5x - 7 = 8x + 14 -> 5x = 8x + 21 -> -3x = 21 -> x = -7
answer asap!!!!!!!!!!!!!!!!!!!
Answer:
Step-by-step explanation:
This is a reflection over x-axis.
(3.6 X 102)(1.9 x 10-4)
Multiply write the answer in scientific notation
Question 2
(4.7. x 10)(2.0 x 10
Answer:
1.)5508 2.)5600 x^2
Step-by-step explanation:
A deer travels at the rate of 42 miles in 6 hours. How many miles does it travel in one hour?
Answer:
7
Step-by-step explanation:
42/6 = 7
The price of a smart watch is $109.99 and the sales tax is 6.5%. What is the total cost of the smart watch? Enter your answer in the box.
Answer:
$117.14
Step-by-step explanation:
A "sales tax" refers to a specific amount of money (according to its percentage) that you need to pay other than the price of the good stated.
So, if the price of the smart watch is $109.99 and the sales tax is 6.5%, you have to know much is 6.5% of $109.99. This can be computed by using the multiplication operation.
109.99 x 6.5% (0.065) = $7.15 (this is the sales tax)
In order to know the total cost of the smart watch, you have to add the sales tax to the price of the smart watch.
$109.99 + $7.15 = $117.14
The total cost of the smart watch is $117.14
Answer:
$117.14
Step-by-step explanation:
I toke the quiz and a k12 student.
please help me so much quick :)
Answer:
22 < x < 74
Step-by-step explanation:
Given 2 sides of a triangle with third side x , then
difference of 2 sides < x < sum of 2 sides , that is
48 - 26 < x < 48 + 26 , so
22 < x < 74
the sides of a triangle are 3 whole 1 by 6 cm 4 whole 2 by 3 cm and 5 whole 2 by 3 cm. Find the total length of the three sides of the triangle
Answer:
[tex]16\frac{2}{3}[/tex]cm
Step-by-step explanation:
Given dimensions:
[tex]3\frac{1}{6}[/tex] cm by [tex]4\frac{2}{3}[/tex]cm by [tex]5\frac{2}{3}[/tex] cm
The total length:
it is the sum of all the sides;
[tex]\frac{19}{6} + \frac{14}{3} + \frac{17}{3}[/tex]
LCM of the denominators = 6 ;
[tex]\frac{38 + 28 + 34}{6}[/tex] = [tex]\frac{100}{6}[/tex]
= [tex]\frac{50}{3}[/tex]
= [tex]16\frac{2}{3}[/tex]cm
A grocery store sells 6 bottles of water for $4 and 18 bottles of water for $10. Which is the better deal and why?
Answer:
18 bottles of water for $10 because each bottle costs 56c compared to 6 bottles of water for $4 at 67c
Step-by-step explanation:
All values rounded to the hundredth
6 bottles of water for $4
4/6=0.67
Each bottle costs 67c
18 bottles of water for $10
10/18=0.56
Each bottle costs 56c
Please help me i will really appreciate it!!!
Answer:
[tex]\displaystyle x=\sqrt{\frac{2U}{k}}[/tex]
The third option is the correct answer.
Step-by-step explanation:
Equation Solving
We have the following equation:
[tex]\displaystyle U=\frac{1}{2}kx^2[/tex]
And it's required to solve it for x.
To solve this equation we need to perform some operations to have the x isolated from any other variables, constants, or operations.
This will be done by conveniently operating on both sides as follows:
Eliminate the denominator by multiplying by 2:
[tex]\displaystyle 2U=kx^2[/tex]
Move k to the other side by dividing by k:
[tex]\displaystyle \frac{2U}{k}=x^2[/tex]
Swapping sides to have x at the left side:
[tex]\displaystyle x^2=\frac{2U}{k}[/tex]
Taking the square root:
[tex]\mathbf{\displaystyle x=\sqrt{\frac{2U}{k}}}[/tex]
The third option is the correct answer.
Write an equation for the nth term of the sequence 12, 5, -2, -9,....
Answer:
n-7
Step-by-step explanation:
when n=12
.: n-7= 12-7=5
when n=5
.:n-7= 5-7= -2
when n=-2
.: -2-7= -9 and so on
The equation that can be used for the nth term of the sequence of 12, 5, -2, -9, and many more will be “n-7”.
What is a sequence?
The sequential ordering of two or more things. two or more objects in orderly succession and orderly chronology. an orderly collection of items or concepts that are connected. something that occurs after another event or series of events.
An enumerated collection of things in mathematics known as a sequence allows repetitions and emphasizes the importance of order. It has members, similar to a set. The sequence's length is expressed in terms of its number of elements.
The word "sequence" refers to a grouping of connected items or occasions, or the succession of their occurrences. A group of numbers in a certain order is called a sequence. There are positions first, second, third, and so on for each phrase in a series.
[tex]n=12\\n-7 = 12-7 =5\\n=5\\n-7 = 5 - 7 = -2\\n=-2\\ -2 -7 = -9[/tex]
Therefore, the equation that can be used for the nth term of the sequence of 12, 5, -2, -9, and many more will be “n-7”.
Learn more about sequence from here:
https://brainly.com/question/21961097
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A brand new car costs $18,000, this price depreciates at 25% every year, how much would the car be after 12 years? 20 years?
Answer:
C(12)=$570.18
C(20)=$57.08
Step-by-step explanation:
Exponential Decay Function
The exponential function is often used to model natural growing or decaying processes, where the change is proportional to the actual quantity.
An exponential decaying function is expressed as:
[tex]C(t)=C_o\cdot(1-r)^t[/tex]
Where:
C(t) is the actual value of the function at time t
Co is the initial value of C at t=0
r is the decaying rate, expressed in decimal
The depreciation of goods is often modeled as an exponential function. The new car costs $18,000 and its price depreciates at r=25%= 0.25 every year.
The depreciation model is:
[tex]C(t)=18,000\cdot(1-0.25)^t[/tex]
Operating:
[tex]C(t)=18,000\cdot 0.75^t[/tex]
At t=12 years, the price will be:
[tex]C(12)=18,000\cdot 0.75^{12}[/tex]
C(12)=$570.18
At t=20 years, the price will be:
[tex]C(20)=18,000\cdot 0.75^{20}[/tex]
C(20)=$57.08
[Congress shall have the power] To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof. Which phrase in this passage signifies implied powers? foregoing Powers Government of the United States necessary and proper Department or Officer
Answer:
the answer is C
Step-by-step explanation:
Answer:
necessary and proper
Can you determine the solution of the |4x−2|≥−6 without solving?
Answer:
|4x-2|">"-6
add 2 to both sides
4x">"-4
divide both sides by 4
x">"-1
the ">" means greater than or equal to, I just don't have that character on my keyboard
Step-by-step explanation:
so the answer is x is greater than or equal to -1
IN: (-∞,-1]
for number line: You would write the number line from -1-10
then, you would put a closed dot on negative one, with an arrow going to the right!
Hope this helped! <3
Lines p, q, r, and s are coplanar. If like p is perpendicular to line q, line q is perpendicular to line r, and line s is parallel to line p, then what conclusion can we make about lines s and r?
A. Lines s and r are parallel
B. Lines s and r are perpendicular
C. Lines s and r are neither parallel nor perpendicular
D. No conclusion can be made
Answer:
A
Step-by-step explanation:
To get the answer here, we simply start looking from what we had initially
From the initial point, we can see that line p and line r are perpendicular
The reason for this is because, line q serves as a point for the two and since q is perpendicular to p and r, then p and r are perpendicular
So since s is parallel to like p and line p
is perpendicular to r, then r and and s must be parallel too
Write the equivalent fraction, the reduced fraction, and the decimal equivalent for 45%. Jenny solved this problem and her work is shown below. What mistake did she make?
Answer:
Equivalent fraction: 45/100
Reduced fraction: 9/20
Decimal equivalent: 0.45
Step-by-step explanation:
Equivalent fraction: percentage is just out of 100. 45% is 45 out of 100, aka 45/100.
Reduced fraction is just simplifying both sides by the GCF. In this case, 5.
Decimal equivalent of a percentage is just dividing the numerator by the denominator, or taking the numerator or the percentage and placing the decimal point forward two digits.
Answer:
45|1000
Step-by-step explanation:
urjrbrbruw9qi1i818
ANSWER PLEASE? I NEED HELP!
Answer:
5x+2=17
x=3
Step-by-step explanation:
Answer:
x=3
Step-by-step explanation:
17-2=15
15/5=3
can somone tell me the interior angle of a polygon with 20 sides
Answer:
162
Step-by-step explanation:
Sum of all the angles of any polygon = (n - 2) * 180
= (20 - 2) *180
= 18 * 180
= 3240
Interior of a polygon with 20 sides = 3240 /20
= 162
how many solutions does 6x - 18 = 4(2x - 3) - (2x +6)
Answer:
x=0
Step-by-step explanation:
6x-18=4(2x-3)-(2x+6)
We move all terms to the left:
6x-18-(4(2x-3)-(2x+6))=0
We calculate terms in parentheses: -(4(2x-3)-(2x+6)), so:
4(2x-3)-(2x+6)
We multiply parentheses
8x-(2x+6)-12
We get rid of parentheses
8x-2x-6-12
We add all the numbers together, and all the variables
6x-18
Back to the equation:
-(6x-18)
We get rid of parentheses
6x-6x+18-18=0
We add all the numbers together, and all the variables
=0
x=0/1
x=0
Answer this question please.
Answer:
x = 11
Step-by-step explanation:
We know that vertical angles (opposite angles when two lines cross), are equal.
2x - 6 = 6x - 50
4x = 44
x = 11
1. What is the domain of the relation
represented by the ordered pairs (-12, 2),
(-6,4), (0, 6), (3, 7), and (9, 9)?
A. {-12, -6,0,3, 9}
B. {2, 4, 6, 7, 9)
C. {3,9}
D. {-12, -6,0, 2, 3, 4, 6, 7,9}
Answer:
A
Step-by-step explanation:
Domain is the x values only
