Answers
Given:
The expression is:
[tex]\dfrac{1}{a+b}+\dfrac{2a}{a^2+b^2}+\dfrac{4a^7+4a^3b^4}{b^8-a^8}[/tex]
To find:
The simplified form of the given expression
Solution:
Formula used:
[tex]x^2-y^2=(x-y)(x+y)[/tex]
We have,
[tex]\dfrac{1}{a+b}+\dfrac{2a}{a^2+b^2}+\dfrac{4a^7+4a^3b^4}{b^8-a^8}[/tex]
It can be written as:
[tex]=\dfrac{1}{a+b}+\dfrac{2a}{a^2+b^2}+\dfrac{4a^3(a^4+b^4)}{(b^4)^2-(a^4)^2}[/tex]
[tex]=\dfrac{1}{a+b}+\dfrac{2a}{a^2+b^2}+\dfrac{4a^3(a^4+b^4)}{(b^4-a^4)(b^4+a^4)}[/tex]
[tex]=\dfrac{1}{a+b}+\dfrac{2a}{a^2+b^2}+\dfrac{4a^3}{(b^2-a^2)(b^2+a^2)}[/tex]
[tex]=\dfrac{1}{a+b}+\dfrac{2a}{a^2+b^2}+\dfrac{4a^3}{(b-a)(b+a)(b^2+a^2)}[/tex]
Taking LCM, we get
[tex]=\dfrac{1(b-a)(b^2+a^2)+2a(b-a)(b+a)+4a^3}{(b-a)(b+a)(b^2+a^2)}[/tex]
[tex]=\dfrac{b^3+a^2b-ab^2-a^3+2a(b^2-a^2)+4a^3}{(b-a)(b+a)(b^2+a^2)}[/tex]
[tex]=\dfrac{b^3+a^2b-ab^2-a^3+2ab^2-2a^3+4a^3}{(b-a)(b+a)(b^2+a^2)}[/tex]
[tex]=\dfrac{b^3+a^2b+ab^2+a^3}{(b-a)(b+a)(b^2+a^2)}[/tex]
Using the grouping method factories the numerator.
[tex]=\dfrac{b(b^2+a^2)+a(b^2+a^2)}{(b-a)(b+a)(b^2+a^2)}[/tex]
[tex]=\dfrac{(b+a)(b^2+a^2)}{(b-a)(b+a)(b^2+a^2)}[/tex]
Cancel out the common factors.
[tex]=\dfrac{1}{b-a}[/tex]
Therefore, the value of the given expression is [tex]\dfrac{1}{b-a}[/tex].
Related Questions
In a class of students, the following data table summarizes how many students have a
brother or a sister. What is the probability that a student chosen randomly from the
class does not have a sister?
Has a brother Does not have a brother
Has a sister
3
12
Does not have a sister
2.
Answers
Answer:
Step-by-step explanation:
First we need to calculate how many total students there are. Once we have the total we have to divide the number in the category that we want by the total number of students to calculate the probability of randomly getting a student in that category.
Total Students = 3 + 12 + 2 + 7 = 24 students
Probabilities:
does not have a sister = [tex]\frac{2 + 7}{24}[/tex] = 0.375 or 37.5% (multiply decimal by 100 to get percentage.
Has a brother = [tex]\frac{3 + 2}{24}[/tex] = 0.208 or 20.8%
Does not have a brother = [tex]\frac{12 + 7}{24}[/tex] = 0.7916 or 79.2%
Has a sister = [tex]\frac{3 + 12}{24}[/tex] = 0.625 or 62.5%
65• 45• work out the size of angle
Answers
Answer:
x = 70
Step-by-step explanation:
The sum of the angles of a triangle add to 180
65+45+x = 180
Combine like terms
110+x = 180
Subtract 110 from each side
x = 180-110
x = 70
On a set of blueprints, the length of a building is 16 inches. The actual building is 85 1/3 feet long. Which of the following is the correct scale?
1:64
1:5
1:32
1:12
Answers
Answer:
1 : 64
Step-by-step explanation:
We require the dimensions to be the same units
1 foot = 12 inches , then
85 [tex]\frac{1}{3}[/tex] feet = [tex]\frac{256}{3}[/tex] × 12 = 256 × 4 = 1024 inches
Scale is then
16 in : 1024 in ( divide both parts by 16 )
= 1 : 64
need help asapppp!!! plz i will give brainliest to the person with the correct answer
Answers
Answer:
Multiply x by 4 and make x negative if it's positive or make x positive if it's negative.
Step-by-step explanation:
Multiplying x by a number greater than 1 will stretch the parabola.
Changing x from positive to negative and vice versa reflects the parabola over the x-axis.
Lisa and Susan are driving to college together. They look at a map to find out how far they have to drive. On the map, Lisa measures the distance to be 4.5 inches. How many miles do they have to drive if the map scale is 1 in. = 40 mi?
A. 10 miles
B. 220 miles
C. 1/10 miles
D. 180 miles
Answers
Answer:
d 180 miles
Step-by-step explanation:
4.5×40= 180 miles
The actual distance to college Lisa and Susan going together is 180 miles.
What is a scale factor?The scale factor defines a variable in a ratio of two different measurements.
Sometimes drawings of large models are represented in a scale factor of a smaller unit.
Given, The map scale is 1 in. = 40 mi.
Lisa and Susan are driving to college together, On the map, Lisa measures the distance to be 4.5 inches.
Therefore,
The actual distance to college is,
= (4.5×40) miles.
= 180 miles.
learn more about scale factors here :
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URGENT PLS HELP!!!
divide £4500 in the ratio 3:7
Answers
Answer:
£1350 & £3150
Step-by-step explanation:
[tex]3x+7x=4500\\10x=4500\\x=450\\\\3(450)=1350\\7(450)=3150[/tex]
Help,anyone can help me do quetion
Answers
Answer:
B2--> the answer is 121 degrees.This is because due to alternate interior angles, the triangle
What is the angle e in the triangle below? (Hint: Use an inverse trigonometric
function.)
A. 22.00
B. 45.30
C. 0.38
D. 68.00
Answers
Answer:
A
Step-by-step explanation:
sin θ=3/8
θ=sin ^{-1}(3/8)
≈22.02 °
Answer:
About 22 degrees
Step-by-step explanation:
Since we are given the opposite side measurement to the angle we wish to find and the hypotenuse measurement, we will want to use sine.
sin(theta)=3/8
theta=arcsin(3/8)
theta=22.024 approximately
Find the area of the shaded region
Answers
Answer:
Area of the shades region = 244.6 ft²
Step-by-step explanation:
Area of the shaded region = Area of the rectangle ABCD- Area of the right angle triangle DEC
Area of the rectangle = Length × width
= 29.8 × 13
= 387.4 ft²
By applying Pythagoras theorem in the right triangle DEC,
(Hypotenuse)² = (leg 1)² + (leg 2)²
CD² = DE² + EC²
(29.8)² = DE² + (28)²
888.04 = DE² + 784
DE² = 888.04 - 784
DE = 10.2 ft
Area of ΔDEC = [tex]\frac{1}{2}(\text{Base})(\text{Height})[/tex]
= [tex]\frac{1}{2}(10.2)(28)[/tex]
= 142.8 ft²
Area of the shades region = 387.4 - 142.8
= 244.6 ft²
What is the solution set of the equation
x + 6 = 2(x + 3) - x?
Answers
Answer:
x = all real numbers
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightDistributive Property
Equality Properties
Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of EqualityAlgebra I
Terms/CoefficientsStep-by-step explanation:
Step 1: Define
Identify
x + 6 = 2(x + 3) - x
Step 2: Solve for x
[Addition Property of Equality] Add x on both sides: 2x + 6 = 2(x + 3)[Distributive Property] Distribute 2: 2x + 6 = 2x + 6Here we see that both sides are identical.
∴ our solution is all real numbers.
I need fast
How many pieces ,each of length 3 ¾ m can be cut from a rope of length 30m?
Answers
Answer:
8
Step-by-step explanation:
Divide 30 by 3.75 to get 8.
Hope that this helps!
A debt payment of $5,500 is due in 27 months. If money is
worth 8.4% p.a. compounded quarterly, what is the equivalent
payment? Now? 15 months? 27 months? 36 months?
Answers
Step-by-step explanation:
A=P(1+0.084) ^n;
=5500(1
The payment now is $4561.7450, 15 months later $5061.28, 27 months later $5500.00595 and 36 months later $5853.83.
Interest rate = 8.4/4 =2.1% = 0.021 every 3 months
The debt of $5500 is due in 27 months.
What is the formula to find compound interest?The formula to find the compound interest is [tex]FV=PV(1+r)^{n}[/tex].
Now, [tex]PV=FV(1+r)^{-n}[/tex].
The payment now:
5500*(1.021)^-9 = $4561.7450
So if $5500 is due in 27 months its current value is $4561.7450.
( b) 15 months from now? 5 quarters
FV(15months)= 4561.75*1.021^5
4561.75*1.021^5 = $5061.28
(c) 27 months from now? 9 quarters
FV(15 months)= $5500 that was given in the question
Verification:
FV(27months)=4561.75*1.021^9
4561.75*1.021^9 = $5500.00595
(d) 36 months from now? 12 quarters
FV(36months)=4561.75*1.021^12
4561.75*1.021^12 = $5853.83
Therefore, the payment now is $4561.7450, 15 months later $5061.28, 27 months later $5500.00595 and 36 months later $5853.83.
To learn more about compound interest visit:
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Please help me with this asap.
Answers
Answer:
80
Step-by-step explanation:
10*8=80
Answer:
Step-by-step explanation:
to find height use pythagoras theorem
height be x
a^2+b^2=c^2
x^2+8^2=10^2
x^2+64=100
xx^2=100-64
x^2=36
x=[tex]\sqrt{36[/tex]
x=6
area of the triangle =base*height/2
=8*6/2
=48/2
=24 square units.
Surface area.. please answer
Answers
[tex] \boxed{ \mathfrak{\huge \: \underline{ \underline{A nswer} }: }}[/tex]
Given terms :
[tex]\hookrightarrow \: length = 8 \: in[/tex]
[tex]\hookrightarrow \: width = 2 \: in[/tex]
[tex] \hookrightarrow \: height = 12 \: in[/tex]
we know,
[tex] \large\boxed{ \mathrm{surface \: \: area \: (a) = 2(lw + wh + lh)}}[/tex]
[tex]\hookrightarrow \mathrm{\: a = 2 \times \bigg [(8 \times 2) + (2 \times 12) + (8 \times 12) \biggr]}[/tex]
[tex]\hookrightarrow a = 2 \times \biggr[16 + 24 + 96 \biggr][/tex]
[tex]\hookrightarrow \: a = \bigg [\: 2 \times 136\bigg ][/tex]
[tex]\hookrightarrow \large \boxed{ \mathrm{a = 272 \: \: in {}^{3} }}[/tex]
____________________________________
[tex] \mathrm{✌TeeNForeveR✌}[/tex]
Brainliest! How can I solve this?
Answers
Please help test tomorrow everything you need is in the photo below
Answers
Answer:
yes
Step-by-step explanation:
6/120 is close to 0, 1/2 or 1
Answers
Answer:
0
Step-by-step explanation:
duh
pleaseee fast help me......................prove that the valueof x obtained by sloving 7^x+1/7^x=49*1/49 also satisfy 2^x-2^2=2^-2-2^-x
Answers
Find the circumference for the following circle. use 22/7 for pi.
A.86.6 m
B.33 m
C.8.4 m
D.16.5 m
Answers
Answer:
D.16.5m
because circumference=21/7×5.25
=16.5m
PLS HELP ME IM RLLY STUCK I WILL GIVE BRAINLIEST
Answers
Check the picture below.
WWhat is the difference of the polynomials?
(–2x3y2 + 4x2y3 – 3xy4) – (6x4y – 5x2y3 – y5)hat is the difference of the polynomials?
Answers
-2x3y2 + 4x2y3 - 3xy4 - 6x4y + 5x2y3 + y5
Collect like terms and arrange in order of powers
-6x4y - 2x3y2 + 9x2y3 - 3xy4 + y5
Write in slope-intercept form an equation of the line that passes through the points (-3, 2) and (1, -10).
Answers
Explanation: In the form y=mx+b, m is the slope and b is the y-intercept. To find the slope given 2 points, you can use the method: (y2-y1)/(x2-x1). When you substitute the points into this formula, it becomes:
(2-(-10))/((-3)-1)
=12/-4
=-3
m=-3
Now that you have the value of m, your equation becomes: y=-3x+b. All you need to do now is find b. To do so, you can choose either one of the points to substitute into your equation. I’ll use the first point as an example:
2=-3(-3)+b
2=9+b
b=-7
Now that you have the value of b, you can write in the final equation: y=-3x-7.
Hope that helps シ
\ 4 people are going to a dinner party. Each person at the party will eat 56 pound of salad. If you can only buy whole pounds of salad, how many pounds should you buy?
Answers
Answer:
224 pounds
step by step :
one person eat 56 pound and we have four person so we must say that :
56 × 4 = 224 pounds
help me pleaseee asap
Answers
Answer:
I think the answer is the blue box
Which of the following is not a possible value for a probability? (A p e x)
A. 0.001
B. 1/16
C. 0.82
D. 5/4
Answers
Explanation: step by step process
Someone plz help me!
Answers
Answer:
56.25 cm
Step-by-step explanation:
your teacher wrote the approach already : the law of sines.
in our example we have
sin(28)/27 = sin(102)/c
c×sin(28)/27 = sin(102)
c×sin(28) = 27×sin(102)
c = 27×sin(102)/sin(28) = 27×0.98/0.47 = 56.25 cm
Find all the missing elements:
40°
120°
8
C = [?]º a = [ ] c = [ ]
Answers
Answer:
C=20° a=10.8 c= 4.3
Step-by-step explanation:
first we need to find the value of angle C by subtracting the sum of angle A and B from 180°.
to find c and a we need to use sin rule which states that sinB/b=sinA/a=sinC/c respectively
If A and B are (-2,-2) and (2,-4). Find the coordinates P such that AP=3/7 AB and P lies on the line segment ab
Answers
Answer:
The coordinates of point P are [tex](-\frac{2}{7}, -\frac{20}{7})[/tex].
Step-by-step explanation:
Point P:
The coordinates of point P are (x,y).
AP=3/7 AB
So
[tex]P - A = \frac{3}{7}(B-A)[/tex]
We apply this both for coordinate x and coordinate y.
Coordinate x:
[tex]x - (-2) = \frac{3}{7}(2 - (-2))[/tex]
[tex]x + 2 = \frac{12}{7}[/tex]
[tex]x = \frac{12}{7} - 2 = \frac{12}{7} - \frac{14}{7} = -\frac{2}{7}[/tex]
Coordinate y:
[tex]y - (-2) = \frac{3}{7}(-4 - (-2))[/tex]
[tex]y + 2 = -\frac{6}{7}[/tex]
[tex]y = -\frac{6}{7} - 2 = -\frac{6}{7} - \frac{14}{7} = -\frac{20}{7}[/tex]
The coordinates of point P are [tex](-\frac{2}{7}, -\frac{20}{7})[/tex].
HELP PLEASE!!!! SHOW WORK!!!
7) Kristof wants to build a new rectangular pen for Sven to live in. Kristof measures and finds that the width of the fence is 3x + 4. He also knows that he wants the length of the fence to be 4 times the width.
a) Create an expression that could be used to find the perimeter of the fence.
b) If Kristof knows that the perimeter of the fence is 280 feet, what is the length and width of the fence?
Answers
Answer: 5
Step-by-step explanation:
Find the equation of a line with slope - 5 passing through the point A(-3, 6).
Answers
Answer:
y=5x-9
Step-by-step explanation:
-5(-3)+b=6
15+b=6
B=-9