PLEASE HELP!!!!! How does “rationalizing the denominator” use the form of 1?

Answers

Answer 1
"Rationalizing the denominator" of a number is exactly what its title implies: turning its denominator into a rational number.

For an example, let's take the fraction 1/√2. We call √2 irrational because it cannot be expressed as a ratio of two integers. Oftentimes when we have irrational numbers like √2 in the denominator, it helps to perform a little bit of algebraic manipulation on them to make the denominator rational. Here's where that "form of 1" comes in. Remember that, as a rational number, 1 can be expressed as n/n, where n is any number. We can use this fact to rationalize the denominator of 1/√2 by multiplying it by √2/√2, which is equivalent to 1.

When we multiply the two together, we get:

[tex] \frac{1}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}= \frac{\sqrt{2}}{(\sqrt{2})^2}= \frac{\sqrt{2}}{2} [/tex]

Crucially, multiplying by 1 in the form of √2/√2 keeps the fraction's value the same, since 1 times any number just equals that same number. If we weren't multiplying but some form of 1, we'd change the number entirely, making the whole attempt at rationalizing the denominator pointless.

Related Questions

Number 19. Answer to number 19) the figure shows a game...

Answers

Where is the figure?
Can you show it?

Whats 0.87 repeating as a fraction?

Answers

we let 100x=87.7 and 10x=8.7
then we subtract the to equations 
100x-10x=8.7-87.7
90x=79
x=79/90
 then 0.87777.....=79/90

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the length of a rectangle is twice its width. The area of the rectangle is 32 square feet. what are the length and width? which equation models this problem?
A) 32w = 2w + w
B) w - 2w = 32
C) 2w x w = 32
D) 2w+w = 32

Answers

l = 2w
area = 32
So, it is C.
The answer is C.
The area of the rectangle is its width times length. The width is w and the length is 2w. 

Paul will pay for his new car in 36 monthly payments.if his car loan is for $19,061,then about how much paul will pat each month? Round your answer to the nearest cent if necessary.

Answers

Final answer:

Paul will pay approximately $529.47 each month for his new car.

Explanation:

To calculate how much Paul will pay each month for his new car, we need to divide the total car loan amount by the number of monthly payments. In this case, the car loan is $19,061 and the number of payments is 36. So, the monthly payment can be calculated as:



Monthly payment = Car loan amount / Number of payments



Substituting the given values, we get:



Monthly payment = $19,061 / 36



Running this calculation, we find that Paul will pay approximately $529.47 each month for his new car, rounded to the nearest cent.

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the length of a rectangle is 2 cm less than 7 times the width. the perimeter is 62 cm. find the length and width

Answers

Perimeter = 2(L) + 2(W) Length = 7W - 2 Perimeter = 60 60 = 14W - 4 + 2(W) 60 =16W - 4 64 = 16W Width = 4 Length = 7(4) - 2 Length = 26
I hope this helped!

Factor this expression completely and then place the factors in the proper location on the grid.

ab + 2a + 3b + 6

Answers

Greetings!

Factor by Grouping.
[tex]ab+2a+3b+6[/tex]
[tex]a(b+2)+3(b+2)[/tex]
Find the GCF.
[tex](b+2)(a+3)[/tex]

Hope this helps.
-Benjamin

Answer:

This was a little confusing. I had trouble getting the right answer, but it is all there. It was (a+3) *(b+2) for me.

Thanks BenjaminG123

ramon has a rope that is 25 1/2 feet long. he wants to cut it into 6 pieces that are equal in length. how long will each piece be?

Answers

4.25. cause you didved 25.5 into 6. cause 25 1/2 turns into 25.5 as a decmail and that the way i got

Jessica made 2 pounds of fudge. How many 1/6 -pound servings are in the 2 pounds of fudge?

Answers

12 because 2 divided into 1/6 will be 2 times six because your will flip the reciprocal and you'll get 12

Using the knowledge of fractions, the number of 1/6-pound servings in 2 pounds of fudge is: 12.

What are Fractions?

Fractions are numbers that represent part of a whole number. They have a denominator and a numerator.

The fraction of a pounds servings is given as 1/6.

Therefore, number of 1/6 pound servings in 2 pounds of fudge would be:

2 ÷ 1/6 = 2 × 6/1 = 12.

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What are some prime numbers between 80 and 90?

Answers

83 and 89 are prime numbers between 80 and 90.

How would you solve 5x+3y=12 by graphing it?

Answers

First you must get y by itself and then set it up in slope intercept form: y=mx+b where m is the slope, b is the y intercept, and y is what it equals too
check the picture below.

a "solution" or zero or root, is pretty much just an x-intercept, namely, what is the value of "x" when the graph touches the x-axis.  Keeping in mind that y = 0 at that moment.

A 10 foot ladder reaches a window 8 feet above the ground, how far from the wall is the base of the ladder

Answers

Final answer:

To find the distance from the wall to the base of the ladder, use the Pythagorean theorem. The distance is 6 feet.

Explanation:

To find the distance from the wall to the base of the ladder, we can use the Pythagorean theorem. The ladder, the ground, and the wall form a right triangle. Let's call the distance from the wall to the base of the ladder x. According to the Pythagorean theorem, the square of the ladder's length (10 feet) is equal to the sum of the squares of the distances from the base of the ladder to the wall (x) and the height of the window (8 feet).

x^2 + 8^2 = 10^2

Simplifying the equation, we get:

x^2 + 64 = 100

Subtracting 64 from both sides:

x^2 = 36

Taking the square root of both sides:

x = 6

Therefore, the distance from the wall to the base of the ladder is 6 feet.

Write an equation for a line perpendicular to y = 2 x + 5 y=2x+5 and passing through the point (-6,1)

Answers

I worked out the problem hope it helps

Find the slope of the line.

Answers

gud pic came at least 69 times and ur slope is 3/2

A line passes through (−1, 7) and (2, 10).

Which answer is the equation of the line?

−3x+y=4−3x+y=4

−3x+y=16−3x+y=16

−x+y=8−x+y=8

−x+y=12

Answers

the formula is y=mx+b
To get the slope or m, use this formula
(the second y minus the first y)/(the second x minus the first x)
Now set it up.
(10-7)/(2--1)
3/3=slope is 1.
y=1x+b

Insert one of the points for x and y.
i will do (-1,7)
7=1(-1)+b
7=-1+b
8=b
Insert this into the final equation:
y=1x+8

Try it out. If you're not sure, try both points. If it works, then you set it up correctly.
i  go to k12 and the correct answer is C hope this works :))

What are the coordinates of the vertex of the graph of f(x)=−3|x−1|?

Answers

I think this is the answer

Find the value of the 100th and 1000th term in the sequence { 3,-5,7,-9,...}

Answers

The value of 100th is 201
and the value of 1000th is 2001

Tati suffered severe stroke damage near the rear of both sides of her brain. She is now unable to perceive

Answers

She is currently unfit to see development. The outcome is oxygen hardship to cerebrum tissue, which can have destroying results. The capacity to recuperate from a stroke relies upon the seriousness of the stroke and how rapidly you get restorative consideration. An enormous stroke can be deadly, yet for some individuals encountering a stroke, recuperation is long however conceivable.

1. If you want to create a cone or cylinder by spinning a 2–dimensional shape about an axis of symmetry, what shapes would you need to use?

2. Describe how the volume formulas for a cone, cylinder and pyramid derived. What are the similarities between the volume of a cone and pyramid?

3. What is the relationship between mass, density and volume? Provide and example that involves calculating population density.

4. Manufacturers often alter different packages to save money and to grab customers attention. Explain using an example, how changes in the dimensions of common geometric shapes (prisms, cylinders, cones and spheres) will affect the volume of these shapes.

5. The concept of volume is something you experience everyday in your life. Provide a specific example in which you could solve a real-world application using volume

Answers

1) The answer for both includes a circle. You would rotate the shape around the axis of symmetry, creating an infinite number of 2-d circles stacked on top of each other, creating the cone or cylinder. In the case of a cylinder, the other shape is a rectangle, and in the case of a cone, a right triangle with one line serving as the axis of symmetry.

2) The volume of a cone is the area of the base, multiplied by height, divided by 3. This means that the volume of a cone is 1/3 the volume of a cylinder of equal height and base area. The volume of a cylinder is the area of the base times height. The volume of a pyramid, similar to a cone, is the area of the base (length x width) times the height, divided by three.

3) One definition of density is mass/volume, or mass per unit of volume. For example, if a 5 kg object has a volume of 10 cm³, than the density would be 0.5kg/cm³. In population density, relative to our example, area takes the place of volume and population takes the place of mass. This means that population density is population/area. For example, if 10,000 people occupy an area of 5 square miles, then the population density is 2,000 people/ square mile. 

4) As you are clearly learning in this class, the volume of an object is not only determined by its dimensions, such as base, width, and height, but its shape. Let's review the cone example. If a cone and a cylinder have the same base area and height, the cone has a volume of 1/3 that of the cylinder. 

5) When might you want to use volume in your daily life? Here is a depressing example. Say your corporate job in the Accounting Department has you feeling blue, and your only break is your daily trip to the water cooler. The cups at the cooler are cone-shaped. Not only is this inconvenient because you can't stand them up without getting the rim dirty, but these cups contain 1/3 the water that a perfectly cylindrical cup of the same base/height could provide. You may tell your manager this, and earn yourself some fancy new cylindrical paper cups! 

Answer: if a cone and a cylinder have the same base and same height then the volume of the cone is the volume of the cylinder.

Step-by-step explanation: its right edge 2021

1) One equation of a system of equations is 5x + 2y = 10. What is the second equation such that the system is independent?

A) x = -2/5y + 2

B) y = -5/2x + 15

C) y = 5/2x + 5

D) y = -5/2x + 5


2) What is the equation of y = |x| translated 1 unit up and 6 units to the left?

A) y = |x + 6| + 1

B) y = |x – 1| + 6

C) y = |x – 6| + 1

D) y = |x + 1| + 6

3) Which statement describes the relationship between the graph of the equation
y=|x/2| and the graph of the equation y=|x/2| +4 ?

A) The graph of y=|x/2| was shifted 4 units to the right.


B) The graph of y=|x/2| was shifted 4 units to the left.


C) The graph of y=|x/2| was shifted 4 units up.


D) The graph of y=|x/2| was shifted 4 units down.



Answer all questions and please show work :)

Answers

1. is C.  2. is B. 3. I don't know so sorry
1) You are supposed to solve for y from the given equation also know as slope-intercept form ... which is y = mx + b 

5x + 2y = 10 

Start by subtracting 5x from both sides....

2y = - 5x + 10 

Divide both sides by 2 to get y by its self...

y = - 5/2 + 5  is the answer

2) A) y = |x + 6| + 1 

Because the graph moved 6 to the left and 1 up

3)  The graph of y=|x/2| was shifted 4 units up
   
From y=|x/2| to the graph of the equation y=|x/2| +4

The formula is  f(x) = a ( x - h ) + k 

    k represents y value .... In the coordinate plane y values go up and down..

    h represents x value .... In the coordinate plane x values go right and left 

For example .....   f(x) = a ( x - 5 ) + 4 

 Means that  the graph moved 5 to the right and 4 units up..




Write the following in scientific notation:

1) 12 756 =

2) 116 464 =

3) 142 984 =

4) 1 392 000 =

5) 59 770 000 =

6) 903 500 000 =

7) 2 644 800 000 =


Answers

1) 1.2756 x 10^4

2) 1.16464 x 10^5

3) 1.42984 x 10^5

4) 1.392 x 10^6

5) 5.977 x 10^7

6) 9.035 x 10^8

7) 2.6448 x 10^9



1.2756 x 10⁴
1.16464 x 10⁵
1.42984 x 10⁵
1.392 x 10⁶
5.977 x 10⁷
9.035 x 10⁸
2.6448 x 10⁹
Hope this helpppped!!!! :D

Type the domain and range values for the corresponding variables that make these relations inverses of each other.

Answers

A) -3.8 B) -2.6 C) 1.7 D) 4.4 E) 1

Answer:

The domain and range values are a) -3.8 b) -2.6 C) 1.7 D) 4.4 E) 1.0

Step-by-step explanation:

Consider the provided table:

The domain of the function becomes the range of the inverse function and the range of the function becomes the domain of the inverse function.

Now consider Figure 1:

The x value of table 1 becomes the y value of table 2 and the y value of table 1 becomes the x value of table 2.

Therefore, the domain and range values are a) -3.8 b) -2.6 C) 1.7 D) 4.4 E) 1.0

Which of the following represents 6 x to the 3 fourths power in radical form?

Answers

6 √x^3 - I am not sure how to write 4th root on the computer but the answer is 6 times the 4th root of x to the 3 third power. 

Answer:

[tex]\\ 6x^{\frac{3}{4}} = 6\sqrt[4]{x^{3}}[/tex]

Step-by-step explanation:

When a number is raised to a rational number like [tex]\\ {\frac{3}{4}}[/tex], that is, a fraction, the denominator represents the index of the radical (or commonly known as the root nth of the radical), and the numerator represents the number to which the radicand (the part of the expression inside the radical sign) is about to be raised.

Looking at [tex]\\ 6x^{\frac{3}{4}}[/tex], the index (root) of the radical is 4 and the numerator raises the value of x to the third power, that is,  [tex]\\ x^{3}[/tex]. That explains the answer: six (6) times [tex]\\ \sqrt[4]{x^{3}}[/tex]. The six (6) only multiplies the expression.

Likewise, a radical of index 5, with a radicand [tex]\\ (x + 8)[/tex] raised to the ninth power is represented by  [tex]\\ \sqrt[5]{(x + 8)^{9}}[/tex], which is equivalent to [tex]\\ (x+8)^{\frac{9}{5}}[/tex].

By the way, a particular case is when the index is 2, that is, [tex]\\ x^{\frac{1}{2}}[/tex]. Here, the number 2 is omitted from the radical symbol and is represented by [tex]\sqrt{x}[/tex].

If y varies inversely as x, and y = 10 as x = 7, find y for the x-value of 10.

Answers

I have the same question... Anyone know the answer?

If 5 people share 1 2 pound of jelly beans equally, what amount will each person receive?

Answers

1/10.

Hope this helps!

Which equation shows y = − 12x +4 in standard form?



A. x + 2y = 8

B. 2x + y = 8

C. x + 2y = − 8

D. 2x − y = 8

Answers

Answer: The equation that shows y=-(1/2)x+4 is standard form is x+2y=8

Option A. x+2y=8


Solution:

y=-(1/2)x+4

Standard form of the equation Ax+By=C, with A, B, and C integer numbers

The coefficient of x is a fraction. To simplify the fraction, let's multiply both sides of the equation by 2:

2y=2 [ -(1/2)x+4]

2y=-2(1/2)x+2(4)

2y=-(2/1)(1/2)x+8

2y=-[(2*1)/(1*2)]x+8

2y=-(2/2)x+8

2y=-1x+8

2y=-x+8

The two variables must be on the left of the equation, then adding x both sides of the equation:

2y+x=-x+8+x

x+2y=8

This is you correct answer

Inequalities:

The dance committee hired a DJ for the fall dance. The DJ charges $125 per hour plus $55 for an assistant. The committee wants to keep the total cost under $555. What is the maximum amount of hours the DJ will play at the dance?

Answers

125x + 55 < 555
Okay, subtract 55 from both sides.
125x < 500
Divide by 125 to isolate the x.
125x < 500
/125    /125
x < 4
The school can have the DJ for less than 4 hours, or for 4 hours. Hope that helps.
Final answer:

The maximum amount of hours the DJ will play at the dance is 4 hours.

Explanation:

To find the maximum amount of hours the DJ will play at the dance, we need to set up an inequality. Let x represent the number of hours the DJ will play. The total cost is given by $125 per hour plus $55 for an assistant, so the cost function is C(x) = 125x + 55. We want the total cost to be less than or equal to $555, so the inequality is 125x + 55 ≤ 555.

To solve this inequality, we need to isolate x. First, subtract 55 from both sides to get 125x ≤ 500. Then, divide both sides by 125 to get x ≤ 4. Therefore, the maximum amount of hours the DJ will play at the dance is 4 hours.

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fifteen more than half a number is 9

Answers

 the answer is 15+1/2x+9

The answer is -12 Hope it helps!

If I could get Brainliest it would completely make my day cause I have a HUGE test coming up and I need points for answering questions :) 

Boat takes 3 hours to go 6 miles upstream. it can go 18 miles downstream in the same time. find the rate of the current and the rate of the boat in still water

Answers

Find the MPH of the boat when it goes upstream... 3h / 6m = 1h / 2m
So, 2mph
Speed of the boat - current = 2mph - > x - y = 2
Find the MPH of the boat going downstream... 3h / 18m = 1h / 6m
So, 6mph
x + y = 6
We now have a system of equation, let's solve it!

x - y = 2 -> x = y + 2
x + y = 6  - >  (y + 2) + y = 6  ->  2y + 2 = 6   ->   2y = 4    ->   y = 2
This is the speed of the current!
Subtitute y = 2 in the first equation:
x - 2 = 2
x = 4
This is the speed of the boat
Rate of the current = 2mph
Rate of the boat in still water = 4mph

A small school has 55 students. if 10 of the students are boys what is the ratio of girls to boys

Answers

For every one boy, there is five and a half girls
55-10=45 girls. 45 girls to 10 boys simplifies to 9 girls to 2 boys (the answer is 9:2)

How do you write algebraic expressions to model quantities?

Answers

Let's say you are walking down the street and you see a lemonade stand. One of the workers tell you that if you buy a lemonade you will have to buy 5 times that amount to sit down and a lemonade cost $2. you order 2 lemonades. after grabbing your drink you feel like sitting down. so you go up to the stand and and attempt to purchase a seat.
5($2x2)=x
(THIS IS JUST AN EXAMPLE) 



Algebraic expressions to model quantities

 A businessman produces x Biscuits made up of wheat and rice daily, and another Business person produces y Snacks made up of vegetables daily.There is another person  who manufactures z , drinks daily.Find total number of Biscuit,snack and Drink produced after 8 days.

So, Algebraic Expression will be

   =8 x+8y+8z

As, there are three terms,so we call it Trinomial Expression.

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