The experimental probability of the arrow stopping over section 1 is 7/13, or 0.538.
To calculate this, you need to take the number of times the arrow stopped on section 1 (45) and divide it by the total number of times the arrow was spun (80). This can be expressed as a fraction (45/80), which can be simplified to 7/13. To convert this fraction to a decimal, divide the numerator by the denominator (7/13 = 0.538).
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The steps for drawing a bearing of 065° from a point, X, are shown below. Put the steps in the correct order.
Answer:
The correct order to draw 65 agle would be:
C
A
B
D
suppose the minimum volume of a clown is 60,000 cm3 and the volume of my car is 3 million cm3. 55 is______ the maximum number of clowns that can fit in my car.
a. An upper bound on
b. Not a bound
c. A lower bound on
d. exactly
When the minimum volume of a clown is 60,000 cm³ and the volume of a car is 3 million cm³, 55 is an upper bound on the maximum number of clowns that can fit in a car. The correct answer is Option A.
What are bounds?Bounds are the maximum and minimum limits that are permitted, within which something can or must be performed. Bounds are used to refer to an acceptable range of values that provide safe operation or performance. In mathematics, a set of bounds can define the limits of the amount of things or objects.
The minimum volume of a clown is 60,000 cm³, so to fit 55 clowns in a car we need:
55 clowns × 60,000 cm³/clown = 3,300,000 cm³
This is lower than the volume of the car. Therefore, 55 is an upper bound on the maximum number of clowns that can fit in the car.
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This was an exceptionally dry year for portions of the southwestern United States. Monthly precipitation in Phoenix, Arizona, was recorded in the table and is modeled by y = –0.04088x2 + 0.4485x + 1.862. In what month did Phoenix receive the lowest amount of precipitation? Month (x) Precipitation January 2.27 inches February ? March ? April ? May ? June ? July ? August ? September 2.59 inches October ? November ? December ? Sketch a graph or fill in the table to answer the question. January February November December
the lowest amount of precipitation occurred in February, with a value of approximately 2.32 inches.
Why it is and how to form a graph?
To find the month with the lowest amount of precipitation, we need to find the minimum value of the quadratic equation y = –0.04088x²2 + 0.4485x + 1.862.
Using calculus, we can find the minimum point of the quadratic function by taking its derivative and setting it equal to zero:
y' = -0.08176x + 0.4485
0 = -0.08176x + 0.4485
x = 5.484
This means that the minimum value of the function occurs at x = 5.484. Since x represents the month number (with January being 1), we can conclude that the month with the lowest amount of precipitation is February (the second month in the table).
To verify this, we can plug in x = 2 into the quadratic equation:
y = –0.04088(2)²2 + 0.4485(2) + 1.862
y = 2.31752
Therefore, the lowest amount of precipitation occurred in February, with a value of approximately 2.32 inches.
To graph the function, we can plot the points given in the table and connect them with a smooth curve. Here is a completed table with the missing values:
Month (x) Precipitation
January 1 2.27 inches
February 2 2.32 inches
March 3 2.57 inches
April 4 2.94 inches
May 5 3.43 inches
June 6 3.94 inches
July 7 2.72 inches
August 8 2.86 inches
September 9 2.59 inches
October 10 2.03 inches
November 11 1.46 inches
December 12 1.03 inches
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Answer:
D: December
The given model for precipitation in Phoenix, Arizona is y = –0.04088x2 + 0.4485x + 1.862, where x is the month number (1 for January, 2 for February, and so on) and y is the precipitation in inches. We can use this model to fill in the missing values in the table:
| Month (x) | Precipitation |
|-----------|---------------|
| January | 2.27 inches |
| February | 2.27 inches |
| March | 2.24 inches |
| April | 2.18 inches |
| May | 2.09 inches |
| June | 1.98 inches |
| July | 1.84 inches |
| August | 1.68 inches |
| September | 2.59 inches |
| October | 1.50 inches |
| November | 1.30 inches |
| December | 1.08 inches |
According to the table, Phoenix received the lowest amount of precipitation in **December** with **1.08 inches** of precipitation, so the correct answer is **D. December**.
An online retailer receives 5% off of all the sales made into here website
After answering the prοvided questiοn, we can cοnclude that the 5% expressiοn discοunt is mοst likely intended tο encοurage custοmers tο purchase.
What is expressiοn ?A cοllectiοn οf symbοls, digits, and cοnglοmerates in mathematics that resemble a pattern οr statistical cοrrelatiοn is knοwn as an expressiοn. Real numbers, mutables, and cοmbinatiοns οf the twο are all acceptable types οf expressiοns. A few examples οf mathematical οperatοrs are additiοn, subtractiοn, rapid spread, divisiοn, and expοnentiatiοn. Expressiοns are frequently used in mathematics, geοmetry, and shape. They are used fοr equatiοn sοlving, representing mathematical fοrmulas, and simplifying mathematical relatiοnships.
All purchases made thrοugh the οnline stοre's website appear tο be eligible fοr a 5% discοunt. The custοmer will thus save 5% οff the tοtal cοst οf the purchase with each sale. Fοr instance, if a custοmer spends $100 οn gοοds, they will get a $5 discοunt, lοwering the final cοst tο $95.
It's wοrth nοting that the retailer may chοοse tο sacrifice prοfits in οrder tο οffer this discοunt. Alternatively, they may have negοtiated lοwer prices with their suppliers οr implemented οther cοst-cutting measures tο cοmpensate fοr the cοst οf the discοunt. In any case, the 5% discοunt is mοst likely intended tο encοurage custοmers tο purchase.
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Complete question:
5 to the power of -4
Show using the value x=2 that
5x³ are not equivalent.
20x^6/4x²
and 5x^3 are not equivalent
Answer:
5x³ = 40
20x^6/4x² = 80
100 is not equal to 80
Step-by-step explanation:
5x³ = 5*2^3
= 40
20x^6/4x² = 20*2^6/4*2^2
= 20*64/16
= 80
For what value of x do 2x+3 and 3x-6 have the same value
Answer:
x = 9
Step-by-step explanation:
[tex]2x + 3 = 3x - 6[/tex]
[tex]3 = x - 6[/tex]
[tex]x = 9[/tex]
A tire company finds the lifespan for one brand of its tires is normally distributed with a mean of 47,500 miles and a standard deviation of 3000. If the manufacturer is willing to replace no more than 10% of the tires, what should be the approximate number of miles for a warranty?
The approximate number of miles for a warranty is 41,748.57 miles.
What is Standard Deviation?Standard deviation refers to the distribution's extent of variability or dispersion. It is a measure of how much the data varies from the mean. It is represented by the Greek letter sigma (σ) and is used to calculate the data variability. It is the square root of the variance of a random variable that describes how widely the data points differ from the average value.
What should be the approximate number of miles for a warranty?We have, Mean (μ) = 47,500 miles
Standard deviation (σ) = 3,000 miles
Manufacturer replacement threshold = 10% = 0.1
The number of miles for a warranty would have to be determined if the tire company wanted to replace no more than 10% of the tires.
Suppose x is the number of miles for which a warranty can be offered.
Using the standard normal distribution, we may compute the z-score as follows:
z = (x - μ) / σ
At the 10th percentile of the standard normal distribution, we'll find the z-score. This z-score can be found using the standard normal distribution table or software.
At the 10th percentile, the z-score is -1.28 (from the table).
Now, substitute the given values.
z = (x - μ) / σ
-1.28 = (x - 47,500) / 3,000
Cross-multiplying and solving for x gives us:
x = -1.28 × 3,000 + 47,500x ⇒ 43,660 miles
Therefore, the tire company should offer a warranty for around 43,660 miles (approximated).
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Please help me answer!
As a result, the percentage of adults who selected math is different from the percentage of kids who did.
what is percentage ?As a number out of 100, a percentage is a method to express a proportion or a fraction. It is symbolized by the number %. If there are 25 boys in a class of 100 pupils, for instance, then there are 25% of boys in the class. It is a helpful method to compare quantities and to express changes in values over time.
given
120 80
Total 200
Women Overall Party A Party B
70 60
Overall 130
Therefore, there are 130 ladies in the group.
b) The chart indicates that 70 women plan to support Party A.
Thus, the percentage of adults who selected English was 40% of 48, which is equal to 0.4 times 48 and 19.2 when rounded to the closest whole number.
b) Reeshma is not accurate. The percentage of adults who selected math is 35%, while for children it is 40%, according to the pie chart.
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The complete question is:- complete the two-way table, which shows the voting intentions of a group of men and women. a How many women are in the group?
Men
Party A Party B 120
Total 200
Women 130
Total 380
b How many women intend to vote for Party A?
2 A group of 48 adults are asked what their favourite subject was at school. They can choose from
maths, English and science.
A group of 32 school children are asked the
same question.
A satellite TV company offers two plans. One plan costs $115 plus $30 per month. The other plan costs $60 per month. How many months must Alfia have the plan in order for the first plan to be the better buy?
I need help, what does this mean
Answer:
2125 ft/min
33,000 ft
y = -2125x + 33,000
Step-by-step explanation:
A. -2125 feet per minute. You get this number when you divide 17,000 by 8 (rise over run). You could also use the formula y2-y/x2-x1 with the points (0, 33,000) and (8, 17,000).
B. 33,000 feet is the height of the plane before it starts descending, so it must be the starting value.
C. Plug in the values you got for A and B into the slope formula y = mx + b
y = -2125x + 33,000
a bowl contains 12 balls, of which 6 are blue, 2 are white, and 4 are red. we draw 4 balls at random, without replacement. what is the probability that the 4 balls include each of the three colors?
The probability of drawing 4 balls at random, without replacement, that include each of the three colors is 0.0061.
What is the probability?The bowl contains 12 balls6 of the balls are blue2 of the balls are white 4 of the balls are red
Sample space = [tex]n(S) = ^{12}C_4=\frac{12!}{4! (12 - 4)!} = 495[/tex]
The formula for conditional probability is:
P(A ∩ B) = P(A) × [tex]P(\frac{B}{A})[/tex]
P(A ∩ B) is the probability that both A and B occur
P(A) is the probability of event A occurring
[tex]P(\frac{B}{A})[/tex] is the probability of event B occurring given that A has occurred
The probability of drawing 4 balls at random that include each of the three colors can be calculated as follows:
First, we calculate the probability of drawing one ball of each color
P (Drawing one ball of each color) = [tex]\frac{(^6C_1^2C_1^4C_1)}{^{12}C_3} = \frac{(6X2X4)}{220} = 0.0545[/tex]
Next, we calculate the probability of drawing a fourth ball of any color
P (Drawing a fourth ball of any color) = [tex]\frac{^9C_1}{^9C_1} = 1[/tex]
Finally, we use the formula for conditional probability to find the probability of drawing 4 balls at random that include each of the three colors
P (Drawing 4 balls at random that include each of the three colors) = P (Drawing one ball of each color) × P (Drawing a fourth ball of any color | Drawing one ball of each color) = [tex](0.0545)\frac{1}{9}= 0.0061[/tex]
The probability of drawing 4 balls at random, without replacement, that include each of the three colors is 0.0061.
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In a hostel, 50 students have food enough for 54 days. How many students should be added in the hostel so that the food is enough for only 45 days ?
We need to add 10 students to the hostel so that the food is enough for only 45 days.
How many students should be added in the hostel so that the food is enough for only 45 days? Let's assume that "x" students need to be added to the hostel so that the food is enough for only 45 days.
We know that the total amount of food available is fixed, so the product of the number of students and the number of days that the food will last must also be fixed.
We can use this fact to set up an equation:
50 × 54 = (50 + x) × 45
To solve for x, we can simplify and solve for x:
Expand using FOIL method
50 × 54 = (50 + x)45
50 × 54 = 45 × 50 + 45 × x
2700 = 2250 + 45x
450 = 45x
x = 10
Therefore, number of students to be added is 10.
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Determine the number of factors of the given whole number
The formula for the total number of factors for a given number is given by; Total Number of Factors for N = (a+1) (b+1) (c+1)
If staff salaries were $44,000/month last year and $53,000/month this year, what is the approximate labor cost Increase in percentage?
A: 3%
B: 7%
C: 10%
D: 21%
The approximate labor cost increase in percentage is approximately 21%, which is closest to option D.
To calculate the approximate labor cost increase in percentage, we can use the following formula:
Labor cost increase percentage = ((new labor cost - old labor cost) / old labor cost) x 100%
Plugging in the given values, we get:
Labor cost increase percentage = ((53,000 - 44,000) / 44,000) x 100%
Labor cost increase percentage = (9,000 / 44,000) x 100%
Labor cost increase percentage = 0.2045 x 100%
Labor cost increase percentage ≈ 20.45%
Therefore, the approximate labor cost increase in percentage is approximately 21%, which is closest to option D.
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Find the missing side of each triangle.
Answer:
5) 12.2 6) 2
Step-by-step explanation:
Use Pythagoras for both
Rearrange for x in question 6
list and describe the four scales of measurement. please provide an example of each one. which measures of central tendency apply to each scale of measurement?
Measurement scales refer to the various methods of assigning scores to measurements according to their precision and accuracy. There are four primary scales of measurement; nominal, ordinal, interval, and ratio.
The four measurement scales and their definitions are listed below:
Nominal scale: This is the lowest level of measurement. It is used to name, classify, or identify objects, events, or groups. It has no logical order, and its units cannot be compared or arranged in a sequence.
Examples of the nominal scale include; gender (male or female), marital status (single or married), color (black or white), and so on.
Ordinal scale: This scale includes a more elaborate classification than nominal measurement. It distinguishes between levels of preference or priority, but it does not have a constant distance between the levels.
Examples of the ordinal scale include; rankings (first, second, third, and so on), level of satisfaction (very happy, happy, neutral, unhappy, very unhappy), and so on.
Interval scale: This scale has a constant distance between its units, and its elements can be compared or arranged in a sequence. It does not have a true zero.
Examples of the interval scale include; temperature (measured in Fahrenheit or Celsius), time (measured in seconds, minutes, or hours), and so on.
Ratio scale: This is the highest level of measurement, where it has a constant distance between its units, its elements can be compared or arranged in a sequence, and it has a true zero.
Examples of the ratio scale include; weight, length, height, and so on. Measures of central tendency include; mean, median, and mode. The mean is the average of all the data points in a set, the median is the middle value in a set of data, and the mode is the value that occurs most often. The table below provides an example of each measurement scale and the appropriate measures of central tendency for each. Scales of Measurement Examples Measures of Central Tendency Nominal Scale Gender Mean Ordinal Scale Ranking Median Interval Scale Temperature Mean Ratio Scale Height Mode.
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in part i, you will evaluate an aircraft altimeter data sample from different companies to determine if there are any differences in error. you have the following three options for running the appropriate t-test. option 1 - compare one brand against a hypothesized mean to determine if the brand is statistically different than the assumed norm. in this case, you would have to cite where the hypothesized population mean came from. option 2 - compare one brand against another to see if there are significant differences between them. option 3 - compare either altimeter error of acme or brand x before adjustment and calibration and after adjustment and calibration to see if there are any differences in the means (performance). based on the type of dependent variable data (ratio) and the type of question in the scenario, you will use a t-test. as you know, there are three different types of t-tests you can use: one sample t-test (test for one population mean using excel) two sample t-test (hypothesis test for two population means using excel) paired t-test (compare before and after values for one type using excel)
From the following three options for running the appropriate t-test, Both Options (2) and (3) could be applicable depending on the specific research question being investigated.
The appropriate t-test to use in this scenario is paired t-test. This test compares before and after values for one type using excel. In this scenario, you would be comparing either the altimeter error of ACME or brand x before adjustment and calibration and after adjustment and calibration to see if there are any differences in the means (performance).
Based on the scenario described, Option 1 is not appropriate because there is no hypothesized population mean to compare against. Therefore, we can exclude this option.
Option 2, comparing one brand against another, could be appropriate if we want to investigate if there are any significant differences in altimeter error between brands. This would be a two-sample t-test.
Option 3, comparing the altimeter error of one brand before and after calibration, could also be appropriate to see if the adjustment and calibration had a significant impact on the altimeter's performance. This would be a paired t-test.
This test is the appropriate test to use because the dependent variable data in this scenario is the ratio and the type of question requires a paired t-test. In this scenario, you are comparing the performance of the two brands before and after adjustment and calibration, which requires paired t-test to answer the research question.
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the u.s. department of transportation recently reported that 81.5% of u.s. airline flights arrived on time. find the probability that among 12 randomly selected flights exactly 10 arrive on time.
The probability that among 12 randomly selected flights, exactly 10 arrive on time is 0.1667
How do we calculate the probability?The probability that, among 12 randomly selected flights, exactly 10 arrive on time is 0.1667. This is calculated by using the binomial probability formula:
[tex]P(x) = nCx (p^x) (1-p)^{(n-x)}[/tex]
Where:
n = 12 (number of flights)x = 10 (number of flights arriving on time)p = 0.815 (probability of one flight arriving on time)[tex]P(x) = 12C10 (0.815^{10}) (0.185)^2 = 0.1667[/tex]
The probability that among 12 randomly selected flights, exactly 10 arrive on time is 0.1667
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Smart mugs are the next generations of hot drinks dispensers tha om with built in technology to keep drinks at the perfect temperature for hours on end initial cost of a smart mug was aed 896, because of high demand in market the cost increased by 12% find the new price of the mug
Pls quick its due 30 mins! And also explain with steps pls! it will help alot, also will make brainliest!
88 books cost \$19. 60$19. 60dollar sign, 19, point, 60. Which equation would help determine the cost of 101010 books?
Choose 1 answer:
The calculation for the price of 10 books is as follows: 10 × (19.60/88) = 2.21
We can use a proportion to determine the cost of 10 books based on the cost of 88 books:
88 books cost $19.60
1 book costs 19.60/88 dollars
Now we can use this information to find the cost of 10 books:
10 books cost 10 × (19.60/88) dollars
Simplifying the expression on the right side, we get:
10 books cost $2.21
Therefore, the equation that would help determine the cost of 10 books is: 10 × (19.60/88) = 2.21 The cost of books is often calculated based on the number of books purchased and the price per book. In this problem, we are given the cost of 88 books, and we need to find the cost of 10 books. Using a proportion, we can find the cost of 1 book and then multiply it by 10 to find the cost of 10 books. This method is useful for determining the cost of any number of books, given the cost of a different number of books. Book prices can vary widely depending on factors such as the format, edition, and publisher, so it is important to carefully consider book prices when making purchases.
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PLEASE HURRY!!!!!!!!!!!!
Graph the solution to this inequality on the number line.
−5+x≥−3
Answer: 2
Step-by-step explanation:
To graph the solution to the inequality -5 + x ≥ -3 on the number line, we first need to isolate x.
Adding 5 to both sides of the inequality, we get:
x ≥ 2
This means that any value of x greater than or equal to 2 will satisfy the inequality. To graph this solution on a number line, we draw a closed circle at the point 2 and shade all the points to the right of 2, including the point 2 itself.
The resulting graph looks like this:
------•-------------------------------->
2
The shaded region on the right of 2 represents all the values of x that make the inequality true.
PLEASE HELP MARKING BRAINLEIST JUST ANSWER ASAP AND BE CORRECT
Answer:
180
Step-by-step explanation:
Form an equation
27x+17=180
Solve it for x
x = 6.0recurring
the perimeter is 180
Let me know if it helps
What is the linear equation of a line that goes through (-4, 3) and (0,6)?
Answer:
y=(3/4)x+6
Step-by-step explanation:
Point 1 (-4,3)
Point 2 (0,6)
To obtain the slope, we apply the formula:
m= (6-3) / (0- (-4))
m= 3 / 4
We replace in the equation y=mx+b.
We take any point.
=> (0,6)
y=mx+b
6=(3/4)(0)+b
b=6
Joining all the terms:
y=(3/4)x+6
A bank statement is shown. Fill in the gaps to complete the sentence below. This bank statement shows a profit/ loss of £ Description Sales - Monday Staff wages Sales - Tuesday Supplies - office equipment Sales - Wednesday Rent for office space Customer refund Money in £360.00 £295.00 £415.00 Money out £520.00 £70.00 £400.00 £150.00 Please help
The bank statement shows a loss of 70 euros, considering the profit concept.
How to calculate the profit?From the table presented in this problem, the parameters are given as follows:
Money in.Money out.To calculate profit considering money in and money out, you need to subtract the total money out from the total money in.
Hence the formula for calculating profit is given as follows:
Profit = Money In - Money Out
The total money in is given as follows:
Money In = 360 + 295 + 415 = 1070.
The total money out is given as follows:
Money Out = 520 + 70 + 400 + 150 = 1140.
Hence the profit is given as follows:
Profit = Money In - Money Out
Profit = 1070 - 1140
Profit = -70 (negative profit means that it is a loss).
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twenty percent of americans ages 25 to 74 have high blood pressure. if 16 randomly selected americans ages 25 to 74 are selected, find each probability. a. none will have high blood pressure. b. one-half will have high blood pressure. c. exactly 4 will have high blood pressure.
Then we will get the following odds
a. None will have high blood pressure. Let the probability of having high blood pressure be denoted by P(A) and the probability of not having high blood pressure be denoted by P(A'). Since none will have high blood pressure, it means all the sixteen Americans selected are healthy, and therefore P(A') = 1. Therefore
P(A) = 1 - P(A')= 1 - 1= 0
b. One-half will have high blood pressure. The probability that one-half of the sixteen Americans will have high blood pressure can be found using the binomial distribution formula that is given by the expression
[tex]P(X = r) = (nCr) * p^r * q^{(n-r)}[/tex]
Where
r = 8n = 16 p = 0.2 q = 1 - p = 0.8Therefore
[tex]P(X = 8) = (16C8) * 0.2^8 * 0.8^8= 0.202[/tex]
c. Exactly 4 will have high blood pressure Similarly, the probability that exactly four of the sixteen Americans will have high blood pressure can be found using the binomial distribution formula as follows:
[tex]P(X = r) = (nCr) * p^r * q^{(n-r})[/tex]
Where
r = 4n = 16p = 0.2q = 1 - p = 0.8Therefore
[tex]P(X = 4) = (16C4) * 0.2^4 * 0.8^{12}= 0.236[/tex]
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Vicky paid $350 for 3 dresses and 4 blouses. Each dress cost as much 3/4 as a blouse. What was the cost of each dress?
Answer:
Let's call the cost of a blouse "b" (in dollars). Then, according to the problem, the cost of a dress is 3/4 of the cost of a blouse, or (3/4)b.
We know that Vicky paid $350 for 3 dresses and 4 blouses, so we can set up the equation:
3(3/4)b + 4b = 350
Simplifying this equation, we get:
9/4 b + 4b = 350
Combining like terms, we get:
17/4 b = 350
To solve for b, we can multiply both sides by 4/17:
b = 350 × 4/17 = $82.35 (rounded to two decimal places)
Therefore, the cost of a dress is:
(3/4)b = (3/4)($82.35) = $61.76 (rounded to two decimal places)
So each dress cost $61.76.
A dart is thrown horizontally at a speed of 10 m/ s at the bull’s-eye of a dartboard 2.4 m away, as in the following figure. (a) How far below the intended target does the dart hit? (b) What does your answer tell you about how proficient dart players throw their darts?
(a) The horizontal velocity of the dart remains constant throughout its motion, so it will take 0.24 seconds to reach the target (since distance = speed × time) and the dart will fall a distance of 0.28 m
During this time, the dart will also be accelerating downwards due to gravity at a rate of 9.81 m/s². The vertical distance fallen can be calculated using the formula: distance = ½ × acceleration × time². Thus, the dart will fall a distance of:
distance = ½ × 9.81 m/s² × (0.24 s)² ≈ 0.28 m
Therefore, the dart will hit the board 0.28 meters below the intended target.
(b) This calculation suggests that even highly skilled dart players may not hit the bull's-eye every time, due to the influence of gravity on the dart's trajectory.
However, skilled players can adjust their aim and release the dart with the correct velocity and angle to hit their target consistently. They may also use techniques such as "aiming high" to compensate for the effect of gravity. Overall, this calculation highlights the importance of understanding the physics of projectile motion in the sport of darts.
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14. a) What are the indicated properties in the following figure? b) What is the nature of GENA? c) Suppose MG=2GO. Prove that L is the midpoint of [EM]. d) What does G represent with respect to triangle MEA?
i do not understand how to answer this question
a. Hence proved that the sum of fractions [tex]${\frac{1}{\sqrt{1+\sqrt{2}}}}+{\frac{1}{\sqrt{2+\sqrt{3}}}}+{\frac{1}{\sqrt{3}+\sqrt{4}}}=1$[/tex]
b. The value will be 7 for the expression
⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+ \cdot \cdot \cdot+{\frac{\sqrt{63}-8}{-1}}$[/tex]
What is square root?Square rοοt οf a number is a value, which οn multiplicatiοn by itself, gives the οriginal number. The square rοοt is an inverse methοd οf squaring a number. Hence, squares and square rοοts are related cοncepts.
Suppοse x is the square rοοt οf y, then it is represented as x=√y, οr we can express the same equatiοn as x² = y. Here, ‘√’ is the radical symbοl used tο represent the rοοt οf numbers. The pοsitive number, when multiplied by itself, represents the square οf the number. The square rοοt οf the square οf a pοsitive number gives the οriginal number.
Here,
a. [tex]${\frac{1}{\sqrt{1+\sqrt{2}}}}+{\frac{1}{\sqrt{2+\sqrt{3}}}}+{\frac{1}{\sqrt{3}+\sqrt{4}}}=1$[/tex]
Using (a + b)(a - b) = a² - b²
⇒ [tex]${\frac{1 \cdot \sqrt{1}-\sqrt{2}}{\sqrt{1}+\sqrt{2}\cdot \sqrt{1 }-\sqrt{2}}+{\frac{1 \cdot \sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}\cdot \sqrt{1}-\sqrt{2}}}+{\frac{1 \cdot \sqrt{3}-\sqrt{4}}{\sqrt{3}+\sqrt{4}\cdot \sqrt{3}-\sqrt{4}}}$[/tex]
⇒ [tex]${\frac{ \sqrt{1}-\sqrt{2}}{1-2}+{\frac{ \sqrt{2}-\sqrt{3}}{2-3}+{\frac{\sqrt{3}-\sqrt{4}}{3-4}}$[/tex]
⇒ [tex]${\frac{ \sqrt{1}-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+{\frac{\sqrt{3}-\sqrt{4}}{-1}}$[/tex]
⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+{\frac{\sqrt{3}-2}{-1}}$[/tex]
⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+{\frac{\sqrt{3}-2}{-1}}$[/tex]
⇒ [tex]$ -1+\sqrt{2}}- \sqrt{2}+\sqrt{3}}-{\sqrt{3}+2}$[/tex]
⇒ [tex]$ -1+2}$[/tex]
⇒ 1
a. Hence proved that the sum of fractions [tex]${\frac{1}{\sqrt{1+\sqrt{2}}}}+{\frac{1}{\sqrt{2+\sqrt{3}}}}+{\frac{1}{\sqrt{3}+\sqrt{4}}}=1$[/tex]
B. This will be done with the same process,
⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+ \cdot \cdot \cdot+{\frac{\sqrt{63}-8}{-1}}$[/tex]
⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+ \cdot \cdot \cdot+{\frac{\sqrt{63}-8}{-1}}$[/tex]
⇒ [tex]$ -1+\sqrt{2}}- \sqrt{2}+\sqrt{3}} \cdot \cdot \cdot -{\sqrt{63}+8}$[/tex]
There, will be same roots of every number until - 8
So,
⇒ [tex]$ -1+8}$[/tex]
= 7
b. The value will be 7 for the expression
⇒ [tex]${\frac{ 1-\sqrt{2}}{-1}+{\frac{ \sqrt{2}-\sqrt{3}}{-1}+ \cdot \cdot \cdot+{\frac{\sqrt{63}-8}{-1}}$[/tex]
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