Express (8 - 3sqrt(2))/(4 + 3sqrt(2)) in the form of a + b * sqrt(2) where a and b are integers.
Answers
Answer: \(\boldsymbol{-25+18\sqrt{2}}\)
The expression is in the form \(a+b\sqrt{2}\) where a = -25 and b = 18
==========================================================
Work Shown:
Let \(x = 3\sqrt{2}\) and then square both sides
\(x = 3\sqrt{2}\\\\x^2 = \left(3\sqrt{2}\right)^2\\\\x^2 = \left(3\right)^2*\left(\sqrt{2}\right)^2\\\\x^2 = 9*2\\\\x^2 = 18\\\\\)
We'll use both x and x^2 later on.
---------------------
The denominator is in the form 4+x. We'll multiply top and bottom by 4-x to then use the difference of squares rule. This will allow us to eliminate the square root in the denominator.
\(\frac{8-3\sqrt{2}}{4+3\sqrt{2}}\\\\\frac{8-x}{4+x}\\\\\frac{(8-x)(4-x)}{(4+x)(4-x)}\\\\\frac{32-8x-4x+x^2}{16-x^2} \ \text{ difference of squares rule}\\\\\frac{32-12x+x^2}{16-x^2}\\\\\)
\(\frac{32-12*3\sqrt{2}+18}{16-18} \ \text{ plug in } x = 3\sqrt{2} \text{ and } x^2 = 18\\\\\frac{50-36\sqrt{2}}{-2}\\\\\frac{-2(-25+18\sqrt{2})}{-2}\\\\\boldsymbol{-25+18\sqrt{2}}\)
Therefore,
\(\frac{8-3\sqrt{2}}{4+3\sqrt{2}}=\boldsymbol{-25+18\sqrt{2}}\)
-------------------------------------
Checking the answer:
Use a calculator to find that,
\(\frac{8-3\sqrt{2}}{4+3\sqrt{2}} \approx 0.45584412271571 \\\\-25+18\sqrt{2} \approx 0.45584412271571\)
We get the same decimal approximation, which helps confirm the correct answer.
Or you could use the idea that if M = N, then M-N = 0 to subtract the original expression M and the final result N. You should get 0 or very close to it.
Related Questions
factorise x³-4x²+x+6
Answers
The binomial factors of x³- 4x²+x+6 are (x+2), (x+3), and (x-1).
Using the splitting and grouping the terms:
x³ + 4x² + x - 6
= x³ + 2x² + 2x² + x - 6 [Splitting 4x² = 2x² + 2x²]
= (x³ + 2x²) + (2x² + x - 6)
= x² (x + 2) + (2x² + 4x - 3x - 6)
= x² (x + 2) + [ 2x (x + 2) - 3 (x + 2)]
= x² (x + 2) + (x + 2) (2x - 3)
= (x + 2) ( x² + 2x - 3)
= (x + 2) ( x² + 3x - x - 3)
= (x + 2) [x (x + 3) - 1 (x + 3)]
= (x + 2) (x + 3) (x - 1)
Hence, the binomial factors are (x + 2), (x + 3) and (x - 1)
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Hurry and answer this please I have to have this right or I'll have to be held back.From a group of 5 boys and 3 girls, a boy AND a girl will be selected to attend a conference. In how many ways can the selection be made?
Answers
Answer:
15 ways
Step-by-step explanation:
girl 1 and boy 1
girl 2 and boy 1
girl 3 and boy1
girl 1 and boy 2
girl2 and boy 2
girl 3 and boy 2
girl 1 and boy 3
girl 2 and boy 3
girl 3 and boy 3
girl 1 and boy 4
girl 2 and boy 4
girl 3 and boy 4
girl 1 and boy 5
girl 2 and boy 5
girl 3 and boy 5
PLEASE HELP!!! SHOW ME HOW YOU GOT THE ANSWER
Answers
Answer:
Step-by-step explanation:
Total amount of money = cost of 13 manicures + tips for 13 manicures
= 13x ( ∵ Given cost of 1 manicure = x) + 52 ( given she get 52 in tips)
= 13x + 52
Question 2
State the probability that a randomly selected, normally
distributed value lies between
a) o below the mean and o above the mean (round to
the nearest hundredths)
b) 20 below the mean and 20 above the mean (round
to the nearest hundredths)
Answers
The probability of a randomly selected, normally distributed value lying between 20 below the mean and 20 above the mean is 0.9545 (rounded to the nearest hundredth).
What is the fundamental concept of probability?A number between zero and one represents the probability that an occurrence will take place. An event is a predefined set of random variable outcomes. Only one mutually exclusive event can occur at a time. Exhaustive events encompass or include all possible outcomes.
We can calculate the probabilities of a randomly chosen value falling between different z-scores using the provided standard normal distribution table.
a) The probability of a value being 0 below or above the mean is the same as the probability of a value being -1 to 1 standard deviations from the mean. According to the standard normal distribution table, the probability of a z-score between -1 and 1 is 0.6827. As a result, the probability of a randomly chosen, normally distributed value falling between 0 below and 0 above the mean is 0.6827. (rounded to the nearest hundredth).
b) The probability of a value falling between 20 and 20 standard deviations from the mean is the same as the probability of a value falling between -20/10 and 20/10 standard deviations from the mean (since the standard deviation is 10). According to the standard normal distribution table, the probability of a z-score between -2 and 2 is 0.9545. As a result, the probability of a randomly chosen, normally distributed value falling between 20 below and 20 above the mean is 0.9545. (rounded to the nearest hundredth).
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John took 30 minutes to bicycle to his grandmother’s house, a total of 2.5 kilometers. What was his speed in km/hr?
Answers
Answer:
5(km)/1(hr)
Step-by-step explanation:
You multiply the 30 minutes to make one hour, and with that, you also multiply the 2.5 kilometers to get 5. You multiply the 2.5 because we multiply both sides of an equation.
Uniform line movement.
We have that the data is:
Time (t) = 30 m = 0.5 h
Distance (d)= 2.5 km
Speed (v) = ?
Since it asks us for the speed in Km/h, we make a time conversion, from minutes to hours. Knowing that 1 hr = 60 minutes, then
\(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{30 \not{m}*\left(\frac{1 \ h}{60 \not{m})\right)=0.5 \ h } } \end{gathered}$}}\)
To calculate the speed, divide the distance by the time. We apply the following formula:
\(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{Speed=\frac{Distance}{Time} } \end{gathered}$}}\)
We substitute our data in the formula and solve:
\(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{V=\frac{d}{t} } \end{gathered}$}}\)
\(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{V=\frac{2.5 \ km}{0.5 \ h} } \end{gathered}$}}\)
\(\boxed{\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{V=5 \ km/h} \end{gathered}$} }}\)
Juan's speed was: 5 km/h.The number of students who ride the school bus is 110%of the number of students who walk.How many students ride the school bus
Answers
If the number of students who walk is represented by x, then the number of students who ride the school bus is 110% of x, which can be written as:
1.10x
This is because 110% can be written as 1.10 (since 110% means 110 out of 100).
So, if we know the value of x, we can multiply it by 1.10 to find the number of students who ride the school bus.
However, we don't know the value of x in this problem. Therefore, we cannot find the exact number of students who ride the school bus.
Answer:If the number of students who walk is represented by x, then the number of students who ride the school bus is 110% of x, which can be written as:
1.10x
This is because 110% can be written as 1.10 (since 110% means 110 out of 100).
So, if we know the value of x, we can multiply it by 1.10 to find the number of students who ride the school bus.
However, we don't know the value of x in this problem. Therefore, we cannot find the exact number of students who ride the school bus.
Step-by-step explanation:
Please answer look at photo
Answers
Answer:
Given Kruger National Park in Skukuza coordinate : (-24.98,31.58).
Nelson Mandela Musuem in Mpumalanga coordinate : (-29.82,31.58).
We need to find exactly in between given coordinates.
We know, mid - point formula :
Therefore, will be meeting at (-27.4, 31.58).
Pls help I’ll give brainlist!
Answers
Answer:
C
Step-by-step explanation:
slope of line perpendicular to y+10=4/7(x-28) is 4/7
slope of line perpendicular to above line is -7/4
eq. of line through (-6,-18) with slope -7/4 is
y+18=-7/4(x+6)
plzzzzzzzzz help me on dis
Answers
Answer:
a
Step-by-step explanation:
please help me answer this question thank you
Answers
Answer:
A
Step-by-step explanation:
Amanda ordered 7 gallons of lemonade for the spring picnic. ONE gallon is approximately 3.3 liters.How many liters of lemonade did Amanda order?
Answers
Answer:
23.1 liters
Step-by-step explanation:
7 x 3.3 = 23.1
Answer:23.
Step-by-step explanation:
Hello! Could someone please help explain how to solve this to me please and an answer would be nice! Thanks! :)
Answers
if we notice the tickmarks on the triangle, we can see the all sides are equal, namely is an equilateral triangle, and if all sides are equal, then all interior angles are equal as well.
Let's recall that the sum of all interior angles in a triangle is 180°, now if we divide that by 3, that gives us 60, namely each angle in the equilateral triangle is 60°, that means
\(10x=60\implies x=\cfrac{60}{10}\implies \boxed{x=6} ~\hfill 12y=60\implies y=\cfrac{60}{12}\implies \boxed{y=5}\)
The phone company Splint has a monthly cellular plan where a customer pays a flat monthly fee and then a certain amount of money per minute used on the phone. If a customer uses 380 minutes, the monthly cost will be $173. If the customer uses 570 minutes, the monthly cost will be $249.
A) Find an equation in the form
y
=
m
x
+
b
,
where
x
is the number of monthly minutes used and
y
is the total monthly cost of the Splint plan.
Answer:
y
=
B) Use your equation to find the total monthly cost if 942 minutes are used.
Answer: If 942 minutes are used, the total cost will be
dollars.
Answers
The solution of the given problem of equation comes out to be total cost for 942 minutes is $1044.
What is an equation?The similar symbol (=) is used in arithmetic equations to signify equality between two statements. It is shown that it is possible to compare various numerical factors by applying mathematical algorithms, which have served as expressions of reality. For instance, the equal sign divides the number 12 or even the solution y + 6 = 12 into two separate variables many characters are on either side of this symbol can be calculated. Conflicting meanings for symbols are quite prevalent.
Part A:
Given:
customer uses 380 minutes, the monthly cost will be $173.customer uses 570 minutes, the monthly cost will be $249.To find an equation,
Where x is number of monthly minutes.
and y is total monthly of splint plan.
So, equation is:
\(\rightarrow \text{y} =\text{mx} +\text{b}\)
For the first case:
\(\rightarrow\bold{173 = 380x + b}\)
Second case:
\(\rightarrow\bold{249= 570x + b}\)
Solve for x:
\(\rightarrow{173 - 380\text{x}=249- 570\text{x}\)
\(\rightarrow{-207=-321\)
\(\rightarrow \text{x} =\dfrac{321}{207}\)
\(\rightarrow \text{x} =\dfrac{107}{69}\)
\(\rightarrow \text{x} \thickapprox1.55\)
For value of b
\(\rightarrow 173 = 380(1.55) + \text{b}\)
\(\rightarrow 173 - 589 = \text{b}\)
\(\rightarrow -416 = \text{b}\)
Part B:
\(\rightarrow \text{y} = 942(1.55) - 416\)
\(\rightarrow \text{y} = 1460.1 - 416\)
\(\rightarrow \text{y} \thickapprox1044\)
Therefore, the solution of the given problem of equation comes out to be total cost for 942 minutes is $1044.
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Given k(x) = 8 - x, find k(-4).
Answers
Answer:
k(-4) = 12
Step-by-step explanation:
What is a function?The most simple definition of a function is a set of inputs that have a given set of outputs.
Functions are often written as f(x) = "an expression"
Where "x" is the input and "the expression" results in an output. Whatever the input is you plug into "x" and replace all "x"s with the input
Here, we are given the function k(x) = 8 - x and we want to find k(-4)
This can be rewritten as k(-4) = 8 - x
==> replace all x's with -4
k(-4) = 8 - (-4)
==> remove parenthesis
k(-4) = 8 + 4
==> add 8 and 4
k(-4) = 12
In a game of chance, players spin the pointer of a spinner with six equal-sized sections.
The numbers are 1 4 4 1 2 4 1 Which outcome has a frequency closest to its expected frequency?
Answer choices
1
2
3
4
Answers
Answer:
2
Step-by-step explanation:
add all the numbers and divide by the number of outcome i.e 17/7 = 2.4 and 2.4 is closest to 2.
Im strugling on these question
Answers
Answer and Step-by-step explanation:
How're you struggling with this? Simply look for a point that is inside of the triangle.
The answer is A. (-1, -2)
because this point is INSIDE the triangle.
#teamtrees #PAW (Plant And Water)
An older person is 5 years older than six times the age of a younger person. The sum of their ages is 26. Find their ages.
Answers
Let x be the age of the younger person.
The older person would be 6x +5 (5 years older than 6 times)
Now add to equal the sum:
x + 6x +5 = 26
Combine like terms:
7x +5 = 26
Subtract 5 from both sides:
7x = 21
Divide both sides by 7:
x = 3
Now you can subtract the age of the younger person from the total age:
26-3 = 23
The ages are 3 and 23
Answer:
3 and 23
Step-by-step explanation:
How do I convert 12mi/hr = ? Ft/hr
Answers
Hilaria borrowed $8,000 from her grandfather to pay for college. Five years later, she paid him back the $8,000, plus $1,600 interest. What was the rate of interest (as a percent)? Solve using the simple interest formula.
Answers
\(~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\dotfill&\$1600\\ P=\textit{original amount deposited}\dotfill & \$8000\\ r=rate\to r\%\to \frac{r}{100}\\ t=years\dotfill &5 \end{cases} \\\\\\ 1600=(8000)\left( \frac{r}{100} \right)(5)\implies \cfrac{1600}{(8000)(5)}=\cfrac{r}{100} \\\\\\ \cfrac{1}{25}=\cfrac{r}{100}\implies \cfrac{100}{25}=r\implies \boxed{\stackrel{\%}{4}=r}\)
Use the slope formula to find the slope of the line through the points (2, 10) and (-3,-6).
Answers
Answer:
3.2
Step-by-step explanation:
Your input: find the slope given two points P=(2,10) and Q=(−3,−6).
The slope of a line passing through the two points P=(x1,y1) and Q=(x2,y2) is given by
m=y2−y1/x2−x1
We have that x1=2, y1=10, x2=−3, y2=−6.
Plug the given values into the formula for slope: m=(−6)−(10)/(−3)−(2)=−16/−5=16/5.
Answer: the slope of the line is m=16/5≈3.2.
when trying to find the probability of a white ball begin drone from a bag contains 4 white balls 3 black balls and 5 red balls how many desire outcomes are there
Answers
White ball = 4
Black ball = 3
Red ball = 5
Total possible outcome = 4 + 3 + 5 = 12
Desire outcomes is a white ball, hence, the number of desire outcomes = 4
Final answer
Number of desire outcomes = 4
which of the following is equivalent to x^2 -5x +6
Answers
Hello!
x² - 5x + 6
= (x² - 2x) + (-3x + 6)
= x(x - 2) - 3(x - 2)
= (x - 2)(x - 3)
what is the location of 0,1? i need help heheheh
Answers
Answer:
Free Coins!!!
Quadrant 1 if that's what your talking about
The graph below shows a couple of plotting points, one of them being 0,1.
Find the area of the irregular figure. Round to the nearest hundredth.
Answers
Answer:
137.5 in^2
Step-by-step explanation:
So, this is basically a rectangle and two right-sides triangles.
First the rectangle: Formula is A=wl
Replace the Values: A = 14*7 98 in^2
Then we solve for the two right-sides triangles.
A= 1 /2a h= 1 /4 b ^2 sin (2 θ )
Top trinangle= A = 25.5 in^2 (height 7 in base 7 in)
Bottom Trinagle A = 14 in^2 (height 7 in base 4 in)
Add results:
98+25.5+14 = 137.5 in^2
6.1 Define the term inflation.
Answers
Answer:
the action of inflating something or the condition of being inflated.
"the inflation of a balloon"
ASTRONOMY
(in some theories of cosmology) a very brief exponential expansion of the universe postulated to have interrupted the standard linear expansion shortly after the Big Bang.
2.
ECONOMICS
a general increase in prices and fall in the purchasing value of money.
"policies aimed at controlling inflation"
3. Which of the following are part of the legislative branch? (1 point)
O House of Representatives and U.S. Senate
OU.S. Supreme Court and White House
OU.S. Senate and U.S. Supreme Court
O White House and House of Representatives
Answers
The table below shows Michaels' income and expenditure for the month of
April 2021. Study the table and answer the questions that follow.
Income
Amount
Andont
R15 435.20
Funeral Plan
Salary
Expenses (Variable)
R 200
R2050
Rent
Petrol
Air Time
Food (Groceries)
Clothing
R 500
R 350
R 400
R 700
School Fees
Car Repayment
Car Insurance
Total Expenditures
R600
R4 500
R 317
A
Entertainment
R 900
R 300
Toiletries
B
Money Left Over
1.1.1 Determine the values of A and B.
Use the value of B to decide whether the family is using the money wisely (2)
Answers
What is the difference in the account balances shown below?
Mia: (-$9.00)
Jacob: $3.00
Answers
Answer:
-6
Step-by-step explanation:
Answer:
12
Step-by-step explanation:
b
help please it's due tomorrow
Answers
\( \{ \: \alpha \: , \: \beta , \: a, \: b \}\)
\( \sf \longrightarrow \: No. \: of \: \: subsets = {2}^{n} \)
where, n denotes to number of elements in set .
Since, given set contains 4 elements .
Thus , 2⁴ {2 raise to power 4} .
\( \sf \longrightarrow \: No. \: of \: \: subsets = {2}^{4} \)
\( \sf \longrightarrow \: No. \: of \: \: subsets = 2 \times 2 \times 2 \times 2\)
\( \sf \longrightarrow \: No. \: of \: \: subsets = 4 \times 4\)
\( \sf \longrightarrow \: No. \: of \: \: subsets = 16\)
Therefore, Required subsets are 16.
They are , Namely;
\( \sf \longrightarrow \: subsets \: = \phi \: \{ \alpha \} \{ \beta \} \{ a\} \{ b\} \: \{ \alpha \beta \} \{ \alpha a\} \{ \alpha b\} \{ \beta a\} \{ \beta b\} \: .....\)
_____________________________
Additional Information:-If n is the number of elements in the set then,
No. of subsets possible for this subset is 2^n that's the (2 raise to the power n).
Let's take another example, {1,2}
Here, n = 2
subsets =2^2 =4
Subsets = ϕ, {1}, {2},{1,2}
Note :- every set is a subset of itself i.e. {1,2} and ϕ is a subset of every set
Given negative 7.3 divided by one half, find the quotient.
−14.6
1.825
−3.65
14.6
Answers
The value of the quotient expression given as negative 7.3 divided by one half is -14.6
How to determine the quotient?The quotient expression is given as:
negative 7.3 divided by one half
Rewrite the above expression properly, as follows:
-7.3/0.5
Evaluate the quotient
-14.6
Hence, the value of the quotient expression given as negative 7.3 divided by one half is -14.6
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Answer:
-14.6
Step-by-step explanation:
Treat the one half as a fraction.
Then remember that when you divide by a fraction, you need to multiply by the reciprocal of the fraction, so dividing by half is the same as multiplying by 2. Also, negative times positive is negative.
-7.3 / (1/2) = -7.3 × 2/1 = -7.3 × 2 = -14.6