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Simplify the following:
(2 sqrt(3)+4) (5 sqrt(3)-2)

Factor 2 out of 4+2 sqrt(3) giving 2 (2+sqrt(3)):
2 (2+sqrt(3)) (5 sqrt(3)-2)

(2+sqrt(3)) (5 sqrt(3)-2) = 2 (-2)+2×5 sqrt(3)+sqrt(3) (-2)+sqrt(3)×5 sqrt(3) = -4+10 sqrt(3)-2 sqrt(3)+15 = 11+8 sqrt(3):
Answer: | 
| 2 [11+8 sqrt(3)]

I believe this has something to do with the angle of incidence and reflection.

The angle of elevation formed by her line of sight and her distance from the mirror  is given by :

atan [5 / 3]   = about 59.04°

And this would be the same angle of elevation formed by the distance the mirror is from the top of the post  and its distance to the post....so we have......

tan (59.04)  = h / 6      where h is the post height

[5/3]  = h / 6     cross-multipy

5 * 6 / 3 = h     =  30/ 3  = 10 ft tall

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I'd like to have someone else look at this one......I might be out in left field, here....!!!!!

Here's a slightly easier method that doesn't require a calculator

27^x = 9^4

[3 ^3]^x  = [3^2]^4

[3]^(3x)  = [3]^(80        and since the bases are equal, we can solve for the wxponents

3x = 8     divide both sides by 3

x = 8/3   =  2+ 2/3   =  2.6666......   [6's repeat ]

>   = greater than

<   = less than

Here's a good way to remember this.......think of them as arrowheads........and the arrowheads always point to the smaller number......!!!!!

Are you serious?????.....It would take us 5 hours to answer all of these.....we're here to help.....NOT to do your homework for you!!!!!!

We can use the tangent function here......let h be the height of the cliff  and d be the initial distance from the boat to the base of the cliff

And we have, in the intial observaation:

tan 44  = h / d   →    h  = d tan 44

And, in the second observation, we have

tan 38  = h / [ d + 200 ]    =   [ d tan 44]  / [ d  + 200]     multiply both sides  by [ d + 200]

[d + 200] tan 38  = d tan 44

d tan 38  + 200 tan 38  = d tan 44       rearrange

d tan 38 - d tan 44  = - 200 tan 38     factor out d on the left

d [ tan 38 -  tan 44]  = - 200 tan 38       divide both sides by [ tan 38 - tan 44]

d = -200 tan38 / [ tan 38 - tan 44]   =   about 847.4 m

And the heght of the cliff =  (847.4) tan (44)  = about 818.3 m

27^x = 9⁴

27^x = 6,561

x = log₂₇6,561

x = 2.6666...

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Take the 35 members and subtract the three extra girls. We are left with 32 members, 16 boys and 16 girls. Add the three girls back on and we have 16 boys and 19 girls.

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-3 - 4 = 7

I'm sure if that was what you are looking for, but here you are.

Let the number of boys=B

B + 3, is the number of girls,

B + B + 3=35

2B=32 divide both sides by 2

B=16 number of boys

16 + 3=19 number of girls

15°/1 • pi/180° = 15pi/180 = pi/12 radians

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(15 X Pi)/180=Pi/12 Radians=(1/12)Pi

2^x + 2^(-x)  = 6  rewrite as

2^x +  1 / [2^x)  = 6        multiply through by  2^x

[2^x][2^x] + 1  = 6[2^x]    subtract 6[2^x]  from both sides

[2^x]^2 -+ 1 - 6[2^x]   = 0       let 2^x = a   ..... and we have.....

a^2  + 1 - 6a  = 0     rearrange

a^2 - 6a + 1 = 0

Using the quadratic formula to solve for a, we have  .......a = 3-2 sqrt(2)   or  a = 3 + 2sqrt(2)

Thus  a = 2^x

And we have that

2^x = 3 - sqrt(2)         take the log of both sides

log2^x   = log [3 -2 sqrt(2) ]     and by a log property, we can write

x*log(2)  = log[ 3 - 2sqrt(2)]   divide both sides by log(2)

x = log[3 - 2sqrt(2)] / log(2)  = about -2.5431

In similar fashion, x also equals.....

 log[ 3 + 2sqrt(2)] / log(2)  = about 2.5431

Here's a graph of  both sides of this strange equation  with the intersection points being the solutions.....

https://www.desmos.com/calculator/uomirk7pas

WHY DON'T WRITE IT AS SUCH?!!!!!!!!!!!!!!!!. You mean:

2^x + 2^-x=6???????????????????????????

Solve for x over the real numbers:
2^(-x)+2^x = 6

Simplify and substitute y = 2^x:
2^(-x)+2^x  =  1/2^x+2^x  =  y+1/y  =  6:
y+1/y = 6

Bring y+1/y together using the common denominator y:
(y^2+1)/y = 6

Multiply both sides by y:
y^2+1 = 6 y

Subtract 6 y from both sides:
y^2-6 y+1 = 0

Subtract 1 from both sides:
y^2-6 y = -1

Add 9 to both sides:
y^2-6 y+9 = 8

Write the left hand side as a square:
(y-3)^2 = 8

Take the square root of both sides:
y-3 = 2 sqrt(2) or y-3 = -2 sqrt(2)

Add 3 to both sides:
y = 3+2 sqrt(2) or y-3 = -2 sqrt(2)

Substitute back for y = 2^x:
2^x = 3+2 sqrt(2) or y-3 = -2 sqrt(2)

Take the logarithm base 2 of both sides:
x = (log(3+2 sqrt(2)))/(log(2)) or y-3 = -2 sqrt(2)

Add 3 to both sides:
x = (log(3+2 sqrt(2)))/(log(2)) or y = 3-2 sqrt(2)

Substitute back for y = 2^x:
x = (log(3+2 sqrt(2)))/(log(2)) or 2^x = 3-2 sqrt(2)

Take the logarithm base 2 of both sides:
Answer: | 
| x = (log(3+2 sqrt(2)))/(log(2))                               or x = (log(3-2 sqrt(2)))/(log(2))

The equation deals with variables as exponents. Thank you for trying to help, though.

I'm assuming this is supposed to be :

S=12pi(3V/[16pi])^(2/3)

(A)  exponentiate both sides to the 3/2 power

S^(3/2)  = 12pi ( 3V/[ 16 pi])

S^(3/2)  = (36/16) V

S^(3/2)  = ( 9/4)V

S^(3/2)  = 2.25V

[ S^(3/2)] / 2.25   = V

(1 / 2.25)S^(3/2)   = V

[0.44]S^(3/2)  = V       [the 0.44 is the "rounded" coefficient]

(B)  12pi(3[150]/[16pi])^(2/3)  = about 162.54  sq ft

BTW - this is correct based on my interpretation of the formula......if the "pi"  in the parentheses in the formula was not supposed to be in the denominator, a clearer  form would have been  :

S = S=12pi[ (3/16)Vpi ] ^(2/3).........I did not interpret it in that way.......I interpreted "16 pi" as the  whole denominator

2x + 2 - x=6

x=6 - 2

x=4

This is a good website to help you

https://www.khanacademy.org/math/arithmetic/fractions/mixed_number_mult_div/v/multiplying-fractions-and-mixed-numbers

-2 * -2 = 2

Two negatives = A positive 

1) Since a ticket for a chaperone is less than a ticket for a student, let  x  represent the cost of a chaperone's ticket.

Then, a student's ticket is 2 + 2x.

The cost of 10 chaperone tickets is 10x.

The cost of 76 student tickets is 76(2 + 2x) =  152 + 152x.

Total cost:  10x + (152 + 152x)  =  162x + 152.

Equation:  162x + 152  =  800

                          162x  =  648

                                 x = $4.00  (cost of a ticket for a chaperone)

                         2 + 2x  =  2 + 2(4.00)  =  $10.00  (cost of a ticket for a student)

2)  Let the cost of a Lamborghini = x.

Then, the cost of a Ferrari  =  x - 10,000

3(Ferrari) + 2(Lambo)  =  2,000,000

       3(x - 10000) + 2x  =  2,000,000

         3x - 30,000 + 2x =  2,000,000

                5x - 30,000  =  2,000,000

                              5x  =  2,030,000

                                x  =    $406,000   (cost of a Lamborghini)

                 x - 10,000  =  406,000 - 10,000  =  $396,000  (cost of a Ferrari)

5 1/3 ÷ -1 1/2

Turn both fractions into improper fraction

16/3 ÷ -3/2

Use the reciprocal

16/3 x -2/3

Multiply

-32/9

Since you can't simplify any further, that's your answer :)

1/2 of the area  = 120

1/2  of the base  = 10     ......so.........

A = (1/2)BH

120 = (1/2)(10)(H)

120/5  = H   =  24

And one of the equal sides  = sqrt [ H^2 + (1/2 of the base)^2 ] =  sqrt [24^2 + 10^2]  = sqrt(676)   = 26

So....the prerimeter=   2*26 + 20  = 72 units

the cost of 10 orange is $1. whats the cost to 10 dozen oranges

$1/10=.10 cents-cost per orange

10 X 12=120 oranges

120 X .10=$12 the cost of 10 dozen oranges.

The cost would be $12

This is a Geometric progression: 1, 1.5, 2.25, 3.375, 5.0625...............365th term.

Sum them all up and you get=3.75 X 10^64 pennies=$3.75 X 10^62 dollars.