Answers
Answer:
√
−
49
=
7
i
Explanation:
A square root of a number
n
is a number
x
such that
x
2
=
n
Note that if
x
is a Real number then
x
2
≥
0
.
So any square root of
−
49
is not a Real number.
In order to be able to take square roots of negative numbers, we need Complex numbers.
That's where the mysterious number
i
comes into play. This is called the imaginary unit and has the property:
i
2
=
−
1
So
i
is a square root of
−
1
. Note that
−
i
is also a square root of
−
1
, since:
(
−
i
)
2
=
(
−
1
⋅
i
)
2
=
(
−
1
)
2
⋅
i
2
=
1
⋅
(
−
1
)
=
−
1
Then we find:
(
7
i
)
2
=
7
2
⋅
i
2
=
49
⋅
(
−
1
)
=
−
49
So
7
i
is a square root of
−
49
. Note that
−
7
i
is also a square root of
−
49
.
What do we mean by the square root of
−
49
For positive values of
n
, the square root is usually taken to mean the principal square root
√
n
, which is the positive one.
For negative values of
n
, the square roots are both multiples of
i
, so neither positive nor negative, but we can define:
√
n
=
i
√
−
n
With this definition, the principal square root of
−
49
is:
√
−
49
=
i
√
49
=
7
i
Footnote
The question remains: Where does
i
come from?
It is possible to define Complex numbers formally, as pairs of Real numbers with rules for arithmetic like this:
(
a
,
b
)
+
(
c
,
d
)
=
(
a
+
c
,
b
+
d
)
(
a
,
b
)
⋅
(
c
,
d
)
=
(
a
c
−
b
d
,
a
b
+
c
d
)
These rules for addition and multiplication work as expected with commutativity, distributivity, etc.
Then Real numbers are just Complex numbers of the form
(
a
,
0
)
and we find:
(
0
,
1
)
⋅
(
0
,
1
)
=
(
−
1
,
0
)
That is
(
0
,
1
)
is a square root of
(
−
1
,
0
)
Then we can define
i
=
(
0
,
1
)
and:
(
a
,
b
)
=
a
(
1
,
0
)
+
b
(
0
,
1
)
=
a
+
b
i
Related Questions
What is the relationship between the standard deviation of the sample mean and the population standard deviation?
Answers
The mean of the sample mean ˉX that we have just computed is exactly the mean of the population. The standard deviation of the sample mean ˉX that we have just computed is the standard deviation of the population divided by the square root of the sample size: √10=√20/√2.
Answer:
Step-by-step explanation:
The mean of the sample mean ˉX that we have just computed is exactly the mean of the population. The standard deviation of the sample mean ˉX that we have just computed is the standard deviation of the population divided by the square root of the sample size: √10=√20/√2.
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Step-by-step explanation:
trust me
Apply the order of operations to simplify each expression.
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Answers
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50
Step-y-step explanation:
Answer:
21
Step-by-step explanation:
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Answers
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a.138 degrees
b.42 degrees
Step-by-step explanation:
What is the total distance in meters of a car that travels due west at 50 m/hr west between 12:00 and 4:00 pm?
Answers
Answer:
A train leaves the station at 9:00 pm traveling east at 36 miles per hour. A second train leaves the station at 10:00 pm traveling west at 42 miles per hour. At what time will both trains have traveled the same distance?
------
Let t equal the time that passed right after 10 PM.
42t = 36 (t + 1)
42t = 36t +36
6t = 36
t = 6 hours
Answer: 4 AM ✅
Step-by-step explanation:
The car traveled 200 m distance with a speed of 50 m / hr in 4 hr.
Given that,
To determine the total distance in meters of a car that travels due west at 50 m/hr west between 12:00 and 4:00 pm.
Speed is defined as when an object is in motion, the distance covered by that object per unit of time is called speed.
What is Distance?Distance is defined as the object traveling at a particular speed in time from one point to another.
Here, the car travels due west at 50 m/hr west between 12:00 and 4:00 pm
Time = 12:00 - 4:00 = 4 hours
Speed = 50 m / hr
Distance = 50 * 4
= 200 m
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Help me guys
Answers
A washer and a dryer cost $959 combined. The washer costs $59 more than the dryer. What is the cost of the dryer?
Answers
Answer:
A dryer costs $450.
A washer costs $509.
Step-by-step explanation:
Let:
"washer" = x
"dryer" = y
Set the equations:
x + y = 959
x = y + 59
First, plug in y + 59 for x in the first equation for x:
x + y = 959
(y + 59) + y = 959
Simplify, combine like terms:
y + y + 59 = 959
2y + 59 = 959
Next, isolate the variable, y. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS. PEMDAS is the order of operations, and stands for:
Parenthesis, Exponents (& Roots), Multiplications, Divisions, Additions, Subtractions.
First, subtract 59 from both sides of the equation:
2y + 59 (-59) = 959 (-59)
2y = 959 - 59
2y = 900
Next, divide 2 from both sides of the equation:
(2y)/2 = (900)/2
y = 900/2
y = 450
Next, plug in 450 for y in one of the equations:
x = y + 59
x = (450) + 59
x = 509
Check:
x + y = 959
450 + 509 = 959
959 = 959 (True).
~
Answer:
$450
Step-by-step explanation:
1. First, let's make a system of equations:
[tex]y+59=x,x+y=959[/tex] (y = dryer, x = washer)2. (Solving)
Solve y + 59 = x for y:
[tex]y + 59 = x[/tex] [tex]y + 59-59=x-59[/tex] (Subtract 59 from both sides)[tex]y = x-59[/tex]Substitute x - 59 for y in x + y = 959
[tex]x + y = 959[/tex] [tex]x+x-59=959[/tex] [tex]2x - 59 = 959[/tex] [tex]2x - 59 + 59 = 959 + 59[/tex] (Add 59 to both sides)[tex]2x = 1018[/tex] [tex]\frac{2x}{2} = \frac{1018}{2}[/tex] (divide both sides by 2)[tex]x = 509[/tex]Substitute 509 for x in y = x - 59
[tex]y = x - 59[/tex] [tex]y = 509-59[/tex] [tex]y = 450[/tex]Therefore, the dryer costs $450. (The washer costs $509).
What is the value of 6(2b-4) when b=5?
Answers
Answer:
36
Step-by-step explanation:
6[(2×5)-4]
6(10-4)
6(6)
=36
Answer:
36
Step-by-step explanation:
6(2b-4) where b=5
= 6(2(5)-4)
= 6(10-4)
= 6(6)
= 36
Which of the following represents this statement? 4x-7>3
Answers
Answer:
b
.................................
Answer:
Step-by-step explanation:
The answer is actually a. Hope this helps :)
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Answers
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Step-by-step explanation:
by HGTV byf g gu o
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Answers
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Answer:
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Answers
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Answers
Answer:
i think you should ask the brain
Step-by-step explanation:
Answer:
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Answers
Answer:
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Step-by-step explanation:
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Answers
Answer:
its B
Step-by-step explanation:
beacuse i searched it it helps
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Answers
Answer:
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Distance
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Step-by-step explanation:
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Answers
Answer:
A
Step-by-step explanation:
A. is the only correct solution to this question because you can also see that if we take 8 and divide this by 32 we get 1/4, so we know it is true.
Because 8 is the same as 8/1 -> 8/1 x 1/4 is 8/4 or 2
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Answers
Answer:
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Answers
Answer:
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Step-by-step explanation:
Answer:
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Step-by-step explanation:
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Answers
9514 1404 393
Answer:
D y²/5 -x²/4 = 1
Step-by-step explanation:
The y-intercept is about 2.2, or √5. This point (0, √5) only satisfies the equation of the last choice:
y²/5 -x²/4 = 1
__
Additional comment
The second attachment tells you the foci are located at (0, ±(a²+b²)), where a² and b² are the denominators in the above equation. That is the foci are ...
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This confirms the equation denominators are rightly 5 and 4, with 5 being the denominator of y².
The second attachment has the equation for a hyperbola that opens along the x-axis. Interchange the variables x and y to get the hyperbola shown in this problem statement.
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Answers
Answer:
1.66Step-by-step explanation:
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Answers
Step-by-step explanation:
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The perimeter of a rectangle is 280 meters. The width is 26 meters less than the length. Find the length and width.
Answers
Answer:
The length is 83 and the width is 57
Step-by-step explanation:
The width is 26 meters less than the length, so we'll represent the width with: L-26. Since there are 2 widths to account for, we'll multiply this by 2.
Equation looks like this so far: 2(L-26)
Then, add the 2 lengths and set the expression equal to 280.
Equation: 2(L-26)+2L=280
Distribute the 2 in the first term, making the equation look like:
2L-52+2L=280
Add the 52 to both sides of the equation
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Answers
Please answer the following questions. D Question 1 Morgan won $2,315 at a work raffle. They deposited $1898.30 into their savir What percent of their winnings did they deposit into their savings account? D Question 2
Answers
In a lottery game, a player picks 9 numbers from 1 to 46. If 8 of the 9 numbers match those drawn, the player wins second prize. What is the probability of winning this prize? Be sure to leave your answer as a reduced fraction in order to earn credit.
There is a _____
probability of winning second prize.
Answers
Answer: [tex]\frac{37}{1,101,716,330}\\\\[/tex]
This is the same as writing 37/1101716330
======================================================
Explanation:
Let's say the nine winning numbers are: 1,2,3,4,5,6,7,8,9
Let's further say that if we matched everything from 1 to 8, then we'll win second prize. That ninth value can be anything but 9. So we could have something like one of these examples
1,2,3,4,5,6,7,8,101,2,3,4,5,6,7,8,111,2,3,4,5,6,7,8,12As you can see, there are 46-9 = 37 different choices we can have for that last slot. In other words, there are 37 items in the set {10,11,12,...,45,46} we can pick for that last ninth slot. Note that 46-10+1 = 37. This value will form the numerator of the answer.
------------------------------------------
Now to calculate how many ways there are to select the nine values regardless if we win or lose.
Since order doesn't matter, we use the nCr combination formula.
[tex]_{n} C _{r} = \frac{n!}{r!*(n-r)!}\\\\_{46} C _{9} = \frac{46!}{9!*(46-9)!}\\\\_{46} C _{9} = \frac{46!}{9!*37!}\\\\_{46} C _{9} = \frac{46*45*44*43*42*41*40*39*38*37!}{9!*37!}\\\\[/tex]
[tex]_{46} C _{9} = \frac{46*45*44*43*42*41*40*39*38}{9!} \ \text{ ... the ``37!'' terms cancel}\\\\_{46} C _{9} = \frac{46*45*44*43*42*41*40*39*38}{9*8*7*6*5*4*3*2*1}\\\\_{46} C _{9} = \frac{399,790,821,830,400}{362,880}\\\\_{46} C _{9} = 1,101,716,330\\\\[/tex]
There are slightly over 1 billion ways to select the nine values.
This large value is the denominator of our answer which is [tex]\frac{37}{1,101,716,330}\\\\[/tex]
The 37 up top represents the number of ways to win the second prize we're after, while the massive number in the denominator represents how many ways there are to play the game.
------------------------------------------
Notice that in the jump from step 4 to step 5, shown in the section above, we had the "37!" terms cancel out. What this means is that the prime number 37 is nowhere to be found anywhere in the factorization of 1,101,716,330
Why is this important? Because it tells us that the fraction [tex]\frac{37}{1,101,716,330}\\\\[/tex] cannot be reduced.
Side note: That fraction converts to the decimal form 0.00000003358397 approximately and that converts to the percentage form 0.000003358397%; both of which are very very small numbers. We can see that the odds are definitely stacked against the player.
NO LINKS OR ASSESSMENT!!
Part 1: Graph a function when give the following Domain and Range:
Answers
Answer:
Step-by-step explanation:
This is basically the same thing as the last question.
The domain goes from - 8 to 5. The 5 has an open circle for it's endpoint < does not include the end point
the range is a little tricky. It goes from - 6 to 9. The -6 is has a circle (because it is not included in the answer). I'm not sure how you show that on a graph.
I think the best answer is this: if either the domain or range is separated by < then you must change the one that uses ≤ into < so the endpoints of the domain are in agreement. You cannot have the domain using < when the range uses ≤. They both must be the same. I think the best choice is<
9514 1404 393
Answer:
see attached
Step-by-step explanation:
The function can have virtually any shape between the ends of the domain and range. The function can have any value in the range at x = -8, and can have any value in the domain at y = 9.
The attachment shows two different functions having the required characteristics.
Evaluate 4 x^3 + 4x when x = 3
Answers
Answer:120
Step-by-step explanation:
Answer:
120
Step-by-step explanation:
Given information:
• 4x³ + 4x
• x = 3
To evaluate the expression further, plug in the given variable and simplify:
4x³ + 4x Replace x with 3.
4(3)³ + 4(3) PEMDAS - put 3 to the 3rd power.
4(27) + 4(3) Use the distributive property to get rid of the parentheses.
108 + 12 Add the two values.
120
The simplification of 4x³ + 4x when x = 3 is 120.