Answers
Answer: [tex]2\sqrt{2}+\sqrt{3}\\\\[/tex]
a = 2 and b = 1
=======================================================
Explanation:
Set the expression equal to the given form we want. Then square both sides so we get rid of the outer-most square root
[tex]\sqrt{11+4\sqrt{6}} = a\sqrt{2}+b\sqrt{3}\\\\\left(\sqrt{11+4\sqrt{6}}\right)^2 = \left(a\sqrt{2}+b\sqrt{3}\right)^2\\\\11+4\sqrt{6} = \left(a\sqrt{2}\right)^2+2*a\sqrt{2}*b\sqrt{3}+\left(b\sqrt{3}\right)^2\\\\11+4\sqrt{6} = 2a^2+2ab\sqrt{2*3}+3b^2\\\\11+4\sqrt{6} = 2a^2+3b^2+2ab\sqrt{6}\\\\[/tex]
In the third line, I used the rule that (x+y)^2 = x^2+2xy+y^2
-------------------
At this point, we equate the non-radical and radical terms to get this system of equations
[tex]\begin{cases}11 = 2a^2+3b^2\\ 4\sqrt{6} = 2ab\sqrt{6}\end{cases}[/tex]
The second equation turns into 4 = 2ab when we divide both sides by sqrt(6)
Then 4 = 2ab turns into ab = 2 after dividing both sides by 2.
We're told that a,b are rational numbers. Let's assume that they are integers (which is a subset of the rational numbers).
If so, then we have these four possibilities
a = 2, b = 1a = -2, b = -1a = 1, b = 2a = -1, b = -2If a,b are negative, then you'll find that [tex]a\sqrt{2}+b\sqrt{3}[/tex] overall is negative. But this contradicts that [tex]\sqrt{11+4\sqrt{6}}[/tex] is positive. So a,b must be positive.
Let's assume that a = 1 and b = 2. If so, then,
2a^2+3b^2 = 2(1)^2+3(2)^2 = 14
but we want that result to be 11 instead.
Let's try a = 2 and b = 1
2a^2+3b^2 = 2(2)^2+3(1)^2 = 11
which works out perfectly.
Therefore,
[tex]\sqrt{11+4\sqrt{6}} = 2\sqrt{2}+\sqrt{3}\\\\[/tex]
---------------------------------
Checking the answer:
Use a calculator to find that
[tex]\sqrt{11+4\sqrt{6}} \approx 4.5604779\\\\2\sqrt{2}+\sqrt{3} \approx 4.5604779\\\\[/tex]
both have the same decimal approximation, so this is a fairly informal way to confirm the answer.
Another thing you can do is to take advantage of the idea that if x = y, then x-y = 0
So if you want to see if two things are equal, you subtract them. You should get exactly 0 or something very small (pretty much equal to 0).
Related Questions
A fruit juice recipe uses three ingredients. Two of the ingredients use a ratio of 2 ounces of pineapple juice to 5 ounces of fruit punch. Part A: Grape a line on the coordinates plane to represent this relationship
Part B: Create an equation to represent this relationship. Let x represent ounces of pineapple juice. Let y represent ounces of fruit punch.
Answers
Answer:
(a) See attachment
[tex](b)\ y =2.5x[/tex]
Step-by-step explanation:
Given
[tex]x \to pineapple[/tex]
[tex]y \to fruit[/tex]
[tex]x : y =2 : 5[/tex]
Solving (a): The graph
First, we create an equation from [tex]x : y =2 : 5[/tex]
Express as fraction
[tex]\frac{x }{ y} =\frac{2 }{ 5}[/tex]
Cross multiply
[tex]2y = 5x[/tex]
See attachment for graph
Solving (b): An expression that represents the relationship
In (a), we have:
[tex]2y = 5x[/tex]
Divide by 2
[tex]y = \frac{5}{2}x[/tex]
[tex]y =2.5x[/tex]
PLZ HELP ASAP DUE IN 10 MINS I WILL MARK BRAINLIEST!!!!
Answers
Answer:
15/8
Step-by-step explanation:
Applying
tanθ = Opposite(O)/Adjacent(A)
tanθ = O/A............ Equation 1
From the diagram,
To get the opposite(O), we apply pythagoras theorem
a² = b²+c²
Where c = opposite
17² = 8²+c²
289 = 64+c²
c² = 289-64
c² = 225
c = √225
c = 15
Substitute into equation 1
Therefore,
tanθ = 15/8
Hence the exact value of tanθ is 15/8
Quadrilateral ABCD has coordinates A(3,-5), B(5,-2), C(10,-4), D(8,-7). Quadrilateral is a
Answers
Answer:
Quadrilateral is another word for a shape with four sides.
Step-by-step explanation:
Answer:
Parallelogram
Step-by-step explanation:
AB=CD
AD=BC
BD not equal to AC (Diagonals are not equal)
AB not equal to BC (Adjacent sides are not equal)
Thus the quadrilateral is parallelogram
Brainliest please
Answer the one that’s is blank
Answers
5x + 15 = 15 + 5x
5x = 5x
x = x
?
2. 5x = -15
3. x = -3
Hope it helps!
The function A(b) relates the area of a trapezoid with a given height of 14 and one base length of 5 with the length of its other base. It takes as input the other base value, and returns as output the area of the trapezoid. A(b)=14xb+5/2 Which equation below represents the inverse function B(a), which takes the trapezoid's area as input and returns as output the length of the other base? A. B(a)+a/7+5 B. B(a)=a/5-7 C. B(a)=a/7-5 D. B(a)=a/5+7
Answers
Answer:
[tex]b(a) =\frac{a}{7} -5[/tex]
Step-by-step explanation:
Given
[tex]A(b) = 14 * \frac{b + 5}{2}[/tex]
Required
Determine the inverse function b(a)
Write A(b) as a
[tex]a = 14 * \frac{b + 5}{2}[/tex]
Swap a and b
[tex]b = 14 * \frac{a + 5}{2}[/tex]
[tex]b = 7(a + 5)[/tex]
Divide both sides by 7
[tex]a + 5 = \frac{b}{7}[/tex]
Maka a the subject
[tex]a =\frac{b}{7} -5[/tex]
Swap a and b again
[tex]b =\frac{a}{7} -5[/tex]
Hence:
[tex]b(a) =\frac{a}{7} -5[/tex]
I need help with questions 11 and 12 please!
Answers
Q11: angle EFG = 90°
angle GDH = 90°÷2
= 45°
angle FEG = 90°÷2
= 45°
angle DHG = 180°-45°-45°
= 90°
Q12. (6x-21)°=90°÷2
=45°
6x=45°+22°
=67°
x=67°÷6
=11/1/6°
hope it helps:)))))
Simplify 3a²b³x 4a³b
Answers
Question 4 Multiple Choice Worth 1 points)
(01.02 MC)
Solve for :
2 x -
(x - 2) = 4x
Answers
Answer:
[tex]\small \sf \frac{2}{3} = x \\ [/tex]
Step-by-step explanation:
Solve for x;
2x - ( x - 2 ) = 4x
distribute minus sign through parantheses
2x - x + 2 = 4xcombine like terms
x + 2 = 4xsubtract x from both side
x - x + 2 = 4x - x2 = 3 xDivide each side by 3
[tex] \small \sf \frac{2}{3} = \frac{3x}{3} \\ [/tex]
[tex]\small \sf \frac{2}{3} = x\: \\ [/tex]
The volume of water remaining in a tank after 10 of its original volume (x) is drained out
Answers
Answer:
x - 10
Step-by-step explanation:
The original volume is x, and 10 is removed.
In mathematical terms, this is expressed as x - 10.
Matilda and Lorraine work in the mail room of a large company sorting letters. Matilda has already sorted 50 letters and continues sorting at a rate of 50 letters per hour. Lorraine has already sorted 80 letters and continues sorting at a rate of 40 letters per hour.
Which function can Matilda and Lorraine use to determine the total number of letters they have sorted after x hours? How many letters will they have sorted after 6 hours?
A.
The function that describes the total number of letters sorted by Matilda and Lorraine in x hours is given by f(x) = 100x + 120. Thus, they will have sorted 720 letters in 6 hours.
B.
The function that describes the total number of letters sorted by Matilda and Lorraine in x hours is given by f(x) = 90x + 130. Thus, they will have sorted 670 letters in 6 hours.
C.
The function that describes the total number of letters sorted by Matilda and Lorraine in x hours is given by f(x) = 130x + 90. Thus, they will have sorted 870 letters in 6 hours.
D.
The function that describes the total number of letters sorted by Matilda and Lorraine in x hours is given by f(x) = 90x + 30. Thus, they will have sorted 570 letters in 6 hours.
Answers
Answer:
B
Step-by-step explanation:
40x + 80
50x + 50
add them together
What is the common difference for this arithmetic sequence? 45,39,33,27,21,...
Answers
Answer:
I expect the correct answer
_6
.
Answer:
6
Step-by-step explanation:
because if u minus 39 from 45 u get six and same for the others
Find the range of the graphed function.
Answers
Answer:
-6 ≤ y ≤ 9
Step-by-step explanation:
Answer:it’s d
Step-by-step explanation:
please give the answer if your answer will be correct I promise that will mark as brainliest
Answers
Answer:
4 1/2 : 2 1/4
Divide both sides by 2 1/4
2 : 1
-------------------------------
3 months : 1 year
1 year = 12 months
3 : 12
Divide both sides by 3
1 : 4
4÷1×2 it the answer and questions
Find the unknown length in the picture below please help
Answers
Answer:
x = 12.65
Step-by-step explanation:
The angle between radius and tangent is 90°
Given diameter is 6, so radius = 3.
Then we use Pythagoras theorem
PLEASE HELP!!
(help me find the type of statement)
Answers
The proof of the given statement below.
What is Congruency?Congruent triangles are triangles that have the same size and shape. This means that the corresponding sides are equal and the corresponding angles are equal.
In ΔDEA and ΔCEA
<DEA= <CEA [Vertically opposite angle]
<ADE= <ACE [given]
AD= CA [given]
Thus, by ASA criteria ΔDEA is congruent to ΔCEA.
Hence, AE = BE [ CPCT]
Learn more about this concept there:
https://brainly.com/question/20521780
#SPJ1
Express 5m2 in cm2 please answer fast!
Answers
Answer:
500000 cm2
Step-by-step explanation:
What is the missing value in the last step? 3 6 9 18
Answers
Answer:
D
Step-by-step explanation:
edge 2021
please help, giving points and brainlest !!
Answers
Answer:
(7,-8)
Step-by-step explanation:
it's a reflection off the x-axis correct me if im wrong i can barely see
Answer:
(7,-8)
Step-by-step explanation:
Point A is at (7,8).
the x-axis is the horizontal line.
Point A is 8 units up of the x-axis so the reflection would be down 8 units from the x-axis in the same line.
Hope this helps.
x² + 8x + 16 =
I need the steps and the answer
and the box
Answers
Answer:
The answer is (x+4)
Step-by-step explanation:
Just write the expression as a product with the factor x and 4
x²+2×x×4+4+16
then write the number in the exponential form with an exponent of 2
x²+2×x×4+4²
then use a² +2ab +b² = (a+b)² yo factor the expression which you get (x+4)
A school is ordering chairs for the teacher's lounge and some of the classrooms. Each classroom needs 17 chairs, while the teacher's lounge needs 16.
Let n represent the number of classrooms included in the order and c represent the total number of chairs. Complete the table using the relationship between n and c.
Answers
5515828288777776677
12
Pls help ..
if A =
[4 3] is a singular matrix, find 'a'
[ a 3]
Answers
Answer:
a = 4.
Step-by-step explanation:
We know that:
[tex]A = \left[\begin{array}{ccc}4&3\\a&3\end{array}\right][/tex]
is a singular matrix, now we want to find the value of a.
We know that a matrix is singular if:
Det [A] = 0
Where remember that the determinant of a 2x2 matrix is given by:
[tex]M = \left[\begin{array}{ccc}k_{11&k_{12\\k_{21}&k_{22}\end{array}\right]\\\\det[M] = k_{11}*k_{22} - k_{12}*k_{21][/tex]
Then the determinant of A is:
Det [A] = 4*3 - a*3
and that must be zero, so:
4*3 - a*3 = 0
Now we just need to find the value of a:
12 - a*3 = 0
12 = a*3
12/3 = a
4 = a
So the value of a is 4.
(FIRST CORRECT FOR BRAINIEST) Solve for x
Answers
Answer:
x = 12
Step-by-step explanation:
Angles in a triangle add up to equal 180 degrees
Hence, 10x - 11 + 3x - 2 + 3x + 1 = 180
( note that we just created an equation that we can use to solve for x )
we now solve for x using the equation we created
10x - 11 + 3x - 2 + 3x + 1 = 180
step 1 combine like terms
10 + 3x + 3x = 16x
- 11 - 2 + 1 = -12
we now have 180 = 16x - 12
step 2 add 12 to each side
- 12 + 12 cancels out
180 + 12 = 192
we now have 192 = 16x
step 3 divide each side by 16
16x / 16 = x
192 / 16 = 12
we're left with x = 12
Answer:
x=12
the seniot math wiz already explained it correctly
Select the two values of x that are roots of this equation. 3x - 5 = -2x2
Answers
Answer:
C and A are the answer
Step-by-step explanation:
step by step
Someone help me on this Problem !!
Answers
Answer:
[tex] \frac{1}{2} [/tex]Negative1 divided by 1 = 12 divided by 4 = 2#Correct me if I'm wrong#Hope it's HelpPlease help…………………………
Answers
Answer:
Step-by-step explanation:
angle 2 is also 60 degree (being vertically opposite angles)
angle 3 is also 60 degree (being corresponding angles)
angle 4 is also 60 degree (being corresponding angles ) . sice angle 2 is 60 degree and relation between angle 2 and 4 is corresponding angle , angle 4 will also be 60 degree.
Help for brainliest
Find h.
9 ft
h = [?] ft
Answers
Answer:
[tex]\sqrt{77}[/tex] ft
Step-by-step explanation:
Hi there!
Notice how we can create a right triangle in the cone, where the base is 2 ft (4/2=2), the height is h and the hypotenuse is 9 ft.
Using these values, we can solve for h with the Pythagorean theorem: [tex]a^2+b^2=c^2[/tex] where a and b are the two legs of a right triangle and c is the hypotenuse.
Plug in 2 and h as a and b and 9 as c:
[tex]2^2+h^2=9^2\\4+h^2=81\\h^2=81-4\\h^2=77\\h=\sqrt{77}[/tex]
I hope this helps!
Name the marked angle in 2 different ways.
Answers
Answer:
∠XWV ∠UWV
Hope this helps! :)
Which compound inequality is represented by the graph?
Answers
Answer:
The answer would be [tex]x < -2[/tex] OR [tex]x \ge 4[/tex].
Step-by-step explanation:
Do pay attention to the hollow circle from which [tex]x < -2[/tex] is found; that means the value of x does not include -2, and therefore the < sign is used.
Hope this helped!
hmmmmmmmm
hope it helps
A 3050 foot long road travels directly up a 550 foot tall hill. Select the equation that can be used to solve for the angle of elevation (0) from the bottom of the hill.
Answers
Answer:
d is the answer ✌sin ०=3000/3050
What is the value of y?
16
50
130
164
Answers
Answer:
where is the figure
Step-by-step explanation:
u should upload the figure too
Answer:
C. 130
Step-by-step explanation: Hope it helps ^w^