For this exercise, we must use the binomial theorem. The formula of this theorem is described as follows
[tex](a+b)^n=\sum_{i\mathop{=}0}^nnCi(a^{n-i})(b^i)[/tex]Where nCi represents the combinatory of n between i. For our case,
[tex]a=a^2\text{ y }b=b[/tex]Replacing the values you have in the formula
[tex]\begin{gathered} (a^2+b)^8=\sum_{i\mathop{=}0}^88Ci(a^2)^{8-i}(b^i) \\ =\frac{8!}{0!(8-0)!}(a^2)^{8-0}b^0+\frac{8!}{1!(8-1)!}(a^2)^{8-1}b^1+\frac{8!}{2!(8-2)!}(a^2)^{8-2}b^2+....+\frac{8!}{8!(8-8)!}(a^2)^{8-8}b^8 \\ =\frac{8!}{8!}(a^2)^8+\frac{8!}{1(7)!}(a^2)^7b^1+\frac{8!}{2(6)!}(a^2)^6b^2+....+\frac{8!}{8!}(a^2)^5b^8 \\ =a^{16}+8a^{14}b+28a^{12}b^2+56a^{10}b^3+70a^8b^4+56a^6b^5+28a^4b^6+8a^2b^7+b^8 \end{gathered}[/tex]Thus,the expansion and simplification is as follows
[tex](a^2+b)^8=\placeholder{⬚}+8a^{14}b+28a^{12}b^2+56a^{10}b^3+70a^8b^4+56a^6b^5+28a^4b^6+8a^2b^7+b^8[/tex]Which is equivalent to StartFraction x Superscript 3 Baseline over StartRoot x EndRoot EndFraction?
We need to re-write x into one exponent:
[tex]\begin{gathered} \sqrt{x}\text{ = x}^{\frac{1}{2}} \\ \frac{x^3}{\sqrt{x}}=\text{ }\frac{x^3}{x^{\frac{1}{2}}} \end{gathered}[/tex]Simplify:
[tex]\begin{gathered} The\text{ operation between the exponent is division} \\ To\text{ combine the exponents, we will subtract the 1/2 from 3} \\ \frac{x^3}{x^{\frac{1}{2}}}\text{ = x}^{3-\frac{1}{2}} \end{gathered}[/tex][tex]\begin{gathered} =\text{ x}^{\frac{6-1}{2}} \\ =\text{ x}^{\frac{5}{2}}\text{ \lparen option C\rparen} \end{gathered}[/tex]The points (2,11) and (6,33) form a proportional relationship. Find the slope of the line through the points. Then use the slope to graph the line.
The slope (m) of the line using the given coordinates is 11/2 and the equation of the slope is plotted on the graph.
What do we mean by slope?The value of the steepness or the direction of a line in a coordinate plane is referred to as the slope of a line, also known as the gradient. Given the equation of a line or the coordinates of points situated on a straight line, a slope can be determined using a variety of approaches.A line's steepness can be determined by looking at its slope. The slope is calculated mathematically as "rise over run" (change in y divided by change in x).So, the slope formula is:
m = y₂ - y₁/x₂ - x₁Now, put the values and calculate as follows:
m = 33 - 11/6 - 2m = 22/4m = 11/2So, the slope equation will be:
y = mx + by = 11/2x + bNow, plot the graph of the equation, taking b = 1.
(Refer to the graph attached below)Therefore, the slope (m) of the line using the given coordinates is 11/2 and the equation of the slope is plotted on the graph.
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what is the slope of a passing line through the points(2, 5) and ( o, -4)?
The slope of the line is 9/2 passing line through the points(2, 5) and ( o, -4)
What is slope ?A line's slope is how steep it is moving from LEFT to RIGHT. The slope of a line is the ratio of its climb, or vertical change, to its run, or horizontal change. The slope remains consistent regardless of where you choose for the line's starting and finishing locations (it never changes).
CalculationWe know that the formula to find the slope of a line is
m = (y2 - y1) / (x2 - x1)
Where (x1, y1) and (x2, y2) are the points
The points given are (2, 5) and (0, -4)
Substituting it in the formula
m = (-4 - 5)/ (0 - 2)
So we get
m = -9/ -2
m = 9/2
Therefore, the slope of the line is 9/2.
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two trains leave the same city at the same time, one going east and the other going west. If one train is traveling at 65 miles per hour and the other at 72 miles per hour, how many hours will it take for them to be 822 miles apart
The effect of both trains moving in opposite directions is the same that if one of them was stationary and the other one was moving at 65 + 72 = 137 mph.
Now, we know that distance = speed x time. Let's call the distance d and the time y
We would have:
[tex]d=137t[/tex]Using the distance we're given,
[tex]822=137t[/tex]Solving for t :
[tex]\begin{gathered} 822=137t \\ \rightarrow\frac{822}{137}=t \\ \Rightarrow t=6 \end{gathered}[/tex]It would take 6 hours for them to be 822 miles apart .
I need help question
The domain is all values of x except 1
the range is all values of y < -1
domain (-∞,1)U(1,∞)
range (-∞,-1)
An aquarium is 0.5 feet wide, 1.5 feet tall, and 2 feet long.The bottom is covered with gravel to a height of 3 inches.The tank will be filled with water to 3 inches below the top.How many gallons of water are needed to fill the aquarium?(Use 1 gallon = 0.134 ft.) Ignore any water that might seepinto the layer of gravel. Round to the nearest tenth.
Step 1: Find the base area of the tank.
The base of the tank is a rectangle of length 2 ft and width 0.5 ft.
Hence,
[tex]\text{ the base area of the tank }=2ft\times0.5ft=1ft^2[/tex]Step 2: Find the height of the water in the tank.
The height of the tank = 1.5ft
the height of the gravel = 3in = 3/12 ft = 1/4 ft
The water is 3 inches (= 1/4 ft) below the top of the tank.
Hence, we must have that
[tex]\begin{gathered} x+\frac{1}{4}+\frac{1}{4}=1.5 \\ x+0.25+0.25=1.5 \\ x+0.5=1.5 \\ x=1.5-0.5=1\text{ ft} \end{gathered}[/tex]Therefore, the height of the water in the tank is 1 ft.
Step 3: Find the volume of the water in the tank
[tex]\begin{gathered} \text{ the volume of the water is given by} \\ V=\text{ base area of the tank }\times\text{ height of the water} \\ \text{ Where V is the volume of the water} \end{gathered}[/tex]Therefore,
[tex]V=1ft^2\times1ft=1ft^3[/tex]Since 1 gallon is equivalengt to 0.134 ft³ then
[tex]V=\frac{1}{0.134}\text{gallons }=7.5\text{ gallons}[/tex]Hence, the number of gallons of water are needed to fill the aquarium is 7.5 gallons.
A desk is on sale for $380, which is 20% off the original price. Which equation can be used to determine theamount of money saved, s, in dollars, when purchasing this desk on sale?A s= 0.2(380)BS = (380 = 0.8) - 380Cs= (380 = 0.5) – 380D8 = (380 = 0.8)
Since the desk was 20% off, the price of $380 represents 80% of the original price.
Step 1. Define an expression to find the original price.
To find this original price, we divide the discounted price of 380 by the percentage it represents: 0.8 (we represent 80% in decimal form as 0.8)
[tex]380\div0.8[/tex]That expression represents the original price.
Step 2. To find the amount saved, we subtract the sale price to the original price:
[tex]s=(380\div0.8)-380[/tex]This expression will give as a result the amount saved s.
Answer:
[tex]s=(380\div0.8)-380[/tex]This is homework the answers are (1/2) (2) (0) (4)
Given:
Given a line.
To determine the slope, we use the formula:
Next, we select the points based on the give line:
We plug in what we know:
Therefore, the slope is 2.
the graph of the functiony=f(x)+34can be obtained from the graph ofy=f(x)by which following action?shifting the graph f(x) to the left 34 unitsshifting the graph f(x) to the right 34 unitsshifting the graph f(x) downwards 34 units shifting the graph of f(x) upwards 34 units
Given data:
The given expression is f(x)+34.
The given graph can be obtained by shifting f(x) in the positive direction of y by 34 units.
F(x)=f(x)+34
Thus, the last option is correct.
2y = 14xDirect variationk =Not direct variation
2y = 14x
divide both sides by 2
so,
y = 7x
direct variation and k = 7
(2 pts)9. Arielle took a cash advance of $1500. Her new credit card charges a periodic (daily)interest rate of 0.07%.a) How much interest will Arielle pay for the statement for the month of May whichhas 31 days(2 pts)b) What is the balance that she owes?
Exercise data
$ 1500
Interest rate = 0.07%= 0.07/ 100 = 0.0007
a) How much interest will Arielle pay for the statement for the month of May which
has 31 days?
Simple Interest=P×I×N
where:
P=principle
I=daily interest rate
N=number of days between payments
Interest= 1500 x (0.07/100) x 31 = $ 32.55
another way to write
Interest= 1500 x (0.0007) x 31 = $ 32.55
b)What is the balance that she owes?
Balance = P x ( 1 + I*t) where t= time.
Balance = 1500 x (1 + 0.0007*31)
Balance = 1500 x (1 + 0.217)
Balance = 1500 x (1 .217)
Balance = 1532.55
another way to solve
Balance = P + Interest
Balance = 1500 + 32.55
Balance = 1532.55
For the function, determine whether y varies directly with x. If so, find the constant of variation and write the function rule.Select the correct choice below and, if necessary, fill in the answer box within your choice.O A. Yes, y varies directly with x. The constant of variation is k = and the function rule is(Simplify your answers. Use integers or fractions for any numbers in the expression)No, y does not vary directly with x.
Answer: No, y does not vary directly with x
Explanation:
y varies directly with x when y divided by x is always the same constant. In this case, we get:
x y y/x
2 6 6/2 = 3
4 14 14/4 = 3.5
5 17 17/5 = 3.4
Since 3, 3.5, and 3.4 are distinct, the answer is No, y does not vary directly with x.
If the ratio of m to nis 7 to 6which of the following could be true?
m = 7/3, n = 2 could be true if the m/n ratio is 7:6. Option-E is correct.
Given that,
Which of the following could be true if the m/n ratio is 7:6.
Option are
A. m = 8, n = 7
B. m = 12,n= 14
C. m=14, n=0
D. m = 42, n = 42
E. m = 7/3, n = 2
Comparing two amounts of the same units and determining the ratio tells us how much of one quantity is in the other.
Consider each of the m and n values that are provided.
Option-A
m:n = 8:7 ≠ 7 : 6
Option-B
m:n = 12:14 = 6:7 ≠ 7 : 6
Option-C
m:n = 14:0 ≠ 7 : 6
Option-D
m:n = 42:42 = 1:1 ≠ 7 : 6
Option-E
m:n = 7/3 : 2 = 7 : 6
Thus E is the only true ratio
Therefore, m = 7/3, n = 2 could be true if the m/n ratio is 7:6.
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Excluding hydropower, U.S. power plants used renewable energy sources to generate 105 million megawatthours of electricity in 2007. By 2012, the amount of electricity generated had increased to 219 million megawatthours. Let x represent the time (in years) since 2007 and let y represent the number of megawatt hours (inmillions).Because time x is defined in years since 2007, 2007 corresponds to x = 0 and 2012 corresponds to x = 5. Fnd therate of change (slope) of the use of renewable energy sources between 2007 and 2012.
In order to calculate the rate of change, we just need to divide the difference in megawatt hours generated between 2007 and 2012 by the period of 5 years.
So we have that:
[tex]\text{rate}=\frac{219-105}{5}=\frac{114}{5}=22.8[/tex]So the rate of change is 22.8 millions of megawatt hours generated per year.
What is the equation of the circle whose diameter has endpoints (10,1) and (-8,1)
Since the endpoints of the diameter are (10, 1) and (-8, 1)
Then the center of the circle is the midpoint of these 2 points
We will use the rule of the midpoint to find the center
[tex]M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex][tex]\begin{gathered} M=(\frac{10+-8}{2},\frac{1+1}{2}) \\ \\ M=(\frac{2}{2},\frac{2}{2}) \\ \\ M=(1,1) \end{gathered}[/tex]The center of the circle is (1, 1)
The length of the diameter is the difference between the x-coordinate
A line has a slope of 8 and passes through the point (1,4). What is its equation in slope-intercept form? Show work
The equation of the line in slope intercept form is:
y = 8x - 4
Given, we have slope = 8
Point = (1,4)
Equation in slope intercept form is:
y = mx+c
let the point (1,4) ne (x₁,y₁)
and slope (m) = 8
y - y₁ = m(x-x₁)
y - 4 = 8(x - 1)
y - 4 = 8x - 8
y = 8x - 8 + 4
y = 8x - 4
Hence the equation in slope intercept form is y = 8x-4
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How can you write the expression with a rationalized denominator?See image
Okay, here we have this:
Considering the provided expression, we are going to rationalize the denominator, so we obtain the following:
[tex]\begin{gathered} \frac{\sqrt[3]{3}}{\sqrt[3]{4}} \\ =\frac{\sqrt[3]{3}\cdot\sqrt[3]{4^2}}{\sqrt[3]{4}\cdot\sqrt[3]{4^2}} \\ =\frac{\sqrt[3]{3}\cdot\sqrt[3]{2^4}}{4^{\frac{1}{3}+\frac{2}{3}}} \\ =\frac{\sqrt[3]{3}\cdot2^{\frac{4}{3}}}{2^2} \\ =\frac{\sqrt[3]{3}\sqrt[3]{2}}{2} \\ =\frac{\sqrt[3]{6}}{2} \end{gathered}[/tex]Finally we obtain that the correct answer is the second option.
Choose the greatest multiple of 10 you can multiply by 30 to get close to,
but not go over, 750. Then, write the product.
Answer:
20, 600
Step-by-step explanation:
[tex]20 \times 30=600 \\ \\ 30 \times 30=900[/tex]
Answer 250 cause 250 is a multiple of 10 times that by 30 is 750
Evaluate the piecewise defined function for the given values of x.F(x)= -x -1 for x<-1, -3 for -1≤x<2, √x-2 for x≥2A. f(-3)B. f(-1)C. f(2)D. f(6)
Answer:
[tex]\begin{gathered} f(-3)=2 \\ f(-1)=-3 \\ f(2)=0 \\ f(6)=2 \end{gathered}[/tex]Explanation:
Given the piecewise defined function;
[tex]f(x)=\mleft\{\begin{aligned}-x-1for\rightarrow x<-1_{} \\ -3\text{for} \\ \sqrt[]{x-2}\text{ for}\rightarrow x\ge2\end{aligned}\mright.\rightarrow-1\leq x<2[/tex]a) f(-3)
we want to find the value of f(x) at x=-3.
-3 is less than -1, so it falls within the interval x<-1.
[tex]\begin{gathered} f(x)=-x-1 \\ f(-3)=-(-3)-1 \\ f(-3)=3-1 \\ f(-3)=2 \end{gathered}[/tex]b) f(-1)
-1 falls with the second interval
[tex]-1\leq x\leq2[/tex]For this interval, f(x) is always equal to -3.
[tex]\begin{gathered} f(x)=-3 \\ f(-1)=-3 \end{gathered}[/tex]c) f(2)
2 falls within the last interval.
[tex]\begin{gathered} f(x)=\sqrt[]{x-2} \\ f(2)=\sqrt[]{2-2} \\ f(2)=0 \end{gathered}[/tex]d) f(6)
6 falls within the last interval.
[tex]\begin{gathered} f(x)=\sqrt[]{x-2} \\ f(6)=\sqrt[]{6-2} \\ f(6)=\sqrt[]{4} \\ f(6)=2 \end{gathered}[/tex]Suppose you and a friend are playing a game that involves flipping a fair coin 3 times. Let X = the number of times
that the coin shows heads. You have previously shown that all conditions have been met and that this scenario
describes a binomial setting.
Determine the value of n and p and calculate the mean and standard deviation of X. Round the standard deviation to
three decimal places.
■ n=
■
■
p=
Hx=
0x =
h
Done
The given solutions are:
n = 3
p = 0.5
mean Hx= 1.5
standard deviation 0x = 0.866
How to solve the probabilityThe value of n = 3
The fair coin is said to have been flipped 3 times
P = 0.5
The coin has two faces, the probability of either face would be 1 / 2
= 0.5
Next we have to solve for the mean of x
μ[tex]_{x}[/tex] = n * p
= number * probability
= 0.5 * 3
= 1.5
The standard deviation
σ[tex]_{x}[/tex] = [tex]\sqrt{np(1-p)}[/tex]
= √1.5(1-0.5)
= √1.5*0.5
= √0.75
= 0.866
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Distributive property to solve-4(s-4)
The distributive property can be applied in the following form:
[tex]a(b-c)=a\times b-a\times c[/tex]Now, the given values are: a=-4, b=s and c=4.
Let's replace them in the equation and solve:
[tex]\begin{gathered} -4(s-4)=(-4)\cdot(s)-(-4)\cdot(4) \\ By\text{ the law of signs -}\cdot-=+and-\cdot+=-\text{ thus:} \\ -4(s-4)=-4s+16 \end{gathered}[/tex]Then, the answer is -4s+16
What fraction is represented in the picture below?
1 5/16
Your answer should be above.
Answer:
A
Step-by-step explanation:
3. Identify the zeros of the equation below: 2x^2 + 23 = 14x
Simplify the equation.
[tex]\begin{gathered} 2x^2+23=14x \\ 2x^2-14x+23=0 \end{gathered}[/tex]Determine the zeros of the equation by using quadratic formula.
[tex]\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ =\frac{14\pm\sqrt[\square]{(-14)^2-4\cdot2\cdot23}}{2\cdot2} \\ =\frac{14\pm\sqrt[]{12}}{4} \\ =\frac{2(7\pm\sqrt[]{3})}{4} \\ =\frac{7}{2}\pm\frac{\sqrt[]{3}}{2} \end{gathered}[/tex]THus zeros of the equation is,
[tex]\frac{7}{2}+\frac{\sqrt[]{3}}{2}[/tex]and
[tex]\frac{7}{2}-\frac{\sqrt[]{3}}{2}[/tex]what do you do in the following problem... "Michael is leaning a 12 foot ladder is leaning against the side of a building. The top of the ladder reaches 10 feet up the side of the building. Approximately how far, to the nearest hundredth, is the bottom of the ladder from the base of the building?"
The distance between the bottom of the ladder to the base of the building = 6.63 feet
Explanations:The height of the ladder, l = 12 feet
Height of the wall covered by the ladder, h = 10 feet
Distance between the base of the building and the bottom of the ladder = x
Using the pythagoras theorem:
[tex]\begin{gathered} (\text{Hypotenuse)}^2=(opposite)^2+(Adjacent)^2 \\ l^2=h^2+x^2 \\ 12^2=10^2+x^2 \\ 144=100+x^2 \\ 144-100=x^2 \\ 44=x^2 \\ \sqrt[]{44}=\text{ x} \\ x\text{ = }6.63 \end{gathered}[/tex]The distance between the bottom of the ladder to the base of the building = 6.63 feet
I need help please!!
Ok, well, I think I need to the see previous problem. But from the information given, I think the right choices are:
a) tenth
b) 31.8
c) 32
Find the volume of this cone.Use 3 for TT.V = Tigh3Hint: The radius (1) is1/2 of the diameter.6 ft6 ft-3V ~ [?]ft
ANSWER
[tex]54ft^3[/tex]EXPLANATION
The volume of a cone is given by:
[tex]V=\frac{\pi r^2h}{3}[/tex]where r = radius
h = height
The height of the given cone is 6 ft and its diameter is 6 ft. The radius of a cone is half its diameter, hence its radius is:
[tex]\begin{gathered} r=\frac{6}{2} \\ r=3ft \end{gathered}[/tex]Hence, the volume of the cone is:
[tex]\begin{gathered} V=\frac{3\cdot3^2\cdot6}{3} \\ V=54ft^3 \end{gathered}[/tex]That is the answer.
For triangle XYZ, m∠X = 38°, m∠Y = (5x − 11)°, and m∠Z = (4x − 45)°. Find m∠Y.
m∠Y = 22°
m∠Y = 43°
m∠Y = 99°
m∠Y = 158°
The measure of angle Y from triangle XYZ is 99°. Therefore, option C is the correct answer.
Given that, in ΔXYZ, m∠X = 38°, m∠Y = (5x - 11)°, and m∠Z = (4x - 45)°.
What is angle sum property of a triangle?Angle sum property of triangle states that the sum of interior angles of a triangle is 180°.
Using angle sum property of triangle, we get
m∠X+m∠Y+m∠Z=180°
⇒ 38°+(5x - 11)°+(4x - 45)°=180°
⇒ 9x+38-11-45=180
⇒ 9x-18=180
⇒ 9x=180+18
⇒ 9x=198
⇒ x=22°
So, m∠Y = (5×22 - 11)°
=110-11
= 99°
The measure of angle Y from triangle XYZ is 99°. Therefore, option C is the correct answer.
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The segments in each figure are tangent to the circle. Solve for x.4x -9AB15DС
CIRCLE THEOREM: Two tangents drawn from an external point to a circle are equal.
Thus, AB is a tangent to the circle and also BC is another tangent to the circle, both drawn from point B.
AB = BC;
[tex]\begin{gathered} 4x-19=15 \\ 4x=15+19 \\ 4x=34 \\ x=\frac{34}{4} \\ x=8.5 \end{gathered}[/tex]The value of x is 8.5
determine if the rate 8 cups with 56 Forks in 4 cups with 28 Forks are equivalent
Divde the number of cups by the number of forks
8 cups to 56 forks = 8 /56 = 0.14
4 cupts to 28 forks = 4 /28= 0.14
Both are rates are equivalent
Which is greater, 1/2 or 2/7?
If we have two fractions a/b and c/d:
[tex]\frac{a}{b}(\text{ })\frac{c}{d}[/tex]To determine which one of them is greater we make the cross product:
[tex]ad(\text{ })bc[/tex]We have 3 cases:
- If ad is greater than bc, then a/b is greater than c/d
- If bc is greater than ad, then c/d is greater than a/b
- If ad is equal to bc, then a/b is equal to c/d
For 1/2 and 2/7:
[tex]\begin{gathered} \frac{1}{2}(\text{ })\frac{2}{7} \\ 7\cdot1(_{\square})2\cdot2 \\ 7>4 \\ \Rightarrow\frac{1}{2}>\frac{2}{7} \end{gathered}[/tex]