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33.333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333% (the threes go on forever)

The fraction -4/3 can be written either as -4/3 or 4/-3.

tan is defined to be opposite divided by adjacent or, when graphed, as y / x.

If you write the fraction as -4/3, then the y-value (opposite) is -4 and the x-value (adjacent) is 3. This point is in Quadrant IV. Using the Pytagorean Theorem, the hypotenuse becomes 5. To find the sin, use the formula for sin, which is opposite over hypotenuse:  sin(x) = -4/5.

If you write the fraction as 4/-3, the y-value is 4 and the x-value is -3, placing the angle in Quadrant II. The hypotenuse is again 5 and the sin becomes 4/5.

So, there are two possible answers.

To factor -1/3 out of -12, you will need to be able to answer the question:  "-1/3 times what number will give you -12?"

To answer this you must divide -12 by -1/3:  -12 ÷ -1/3  =  -12 x 3/-1  =  36.

You can then write -12 as  (-1/3)(36):

To factor -1/3x - 12:  =  (-1/3)x + (-1/3)(36)

                                   =  (-1/3)( x + 36 ) 

3^(x+2)=5^x

Taking the log of both sides, we have

log3^(x+2)= log5^x    and b a property of logs, we can write

(x+2) log 3 = x log 5     simplify

x log 3   + 2 log 3  = x log 5      rearrange

2 log 3  = x log 5 - x log 3          factor on the right

2 log 3 = x (log 5 - log 3)           divide both sides by  (log 5 - log 3)

(2 log 3) / (log 5 - log 3)  = x =  about 4.3013

just type in ^(power)

The correct answer to "55+678" is 733

$${\mathtt{55}}{\mathtt{\,\small\textbf+\,}}{\mathtt{678}} = {\mathtt{733}}$$

The correct answer to "two plus two" is 4 

$${\mathtt{2}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}} = {\mathtt{4}}$$

A =  {Galway, Mayo}    ..... these each have an even number of letters

B = {Sligo, Leitrim}      ........each has a prime number of letters..i.e., 5 and 7

"Roscommon" belongs to neither set.......its number of letters - 9 - is neither even, nor prime

Sligo has 5 letters

Galway has 6 letters

May has 4 letters

Roscommon has 9 letters

Leitrim has 7 letters

To answer part a: Which of these names have an even number of letters? List those names in set A:

     A  =  {          }

To answer part b: Which of these names have a prime number of letters? List those names in set B:

     B  =  {         }

If there are no answers that fit, leave the set empty. If there are more than one answer, list all answers, placing commas between answers. 

This equation is solved for Z already.  Z is isolated on one side and everything else is on the other side.

25 - 13f  =  -14

First step:  get the variable term, the f-term, by itself:  

Subtract 25 from both sides:

--->   -13f = -14 - 25

--->   -13f  =  -39

Second step:  divide both sides by -13:

--->   -13f / -13  =  -39 / -13

--->   f  =  3

35/11 is a common fraction.

As a mixed fraction:  35 ÷ 11 = 3, with a remainder of 2.

--->   35/11  =  3 2/11

As a decimal fraction:  35 ÷ 11 = 3.18181818...

If  Π/Π/2  =  (Π/Π)/2  = 1/2

If  Π/Π/2  =  Π/(Π/2)   = 2

I'm going to assume the second answer.

So, I'm going to make Π/Π/2Ψ  =  Π/(Π/2Ψ)  = 2Ψ.

Problem:  2cos(2Ψ)cos²(Ψ) - 2sin(2Ψ)cos(Ψ)(sin(Ψ) 

             =  2[ cos(2Ψ)cos²(Ψ) - sin(2Ψ)cos(Ψ)(sin(Ψ) ]

Since sin(2X) = 2sin(X)cos(X):

             =  2[ cos(2Ψ)cos²(Ψ) - sin(Ψ)cos(Ψ)cos(Ψ)(sin(Ψ) ]

             =  2[ cos(2Ψ)cos²(Ψ) - sin²(Ψ)cos²(Ψ) ]

             =  2cos²(Ψ)[ cos(2Ψ)  - sin²(Ψ) ]

             =   2cos²(Ψ)[ 1 - 2sin²(Ψ) - sin²(Ψ) ]

             =  2cos²(Ψ)[ 1 - 3sin²(Ψ) ] 

From here ... ?

(2/3) - (3/2) / (-4/3)

To divide fractions, multiply by the reciprocal:

(2/3) - (3/2) x (3/-4)

(2/3) - (9/-8)

(2/3) + (9/8)

Common denominator is 24:

(2/3)(8/8) + (9/8)(3/3)

16/24 + 27/24

43/24

1.7916666666...

You can enter that into the calculator just as shown to get the answer.

-4 + -6  =  -10          (a negative plus a negative is a negative)

3(-4)  =  -12             (a positive times a negative is a negative)

-10 / -12  =  5/6       (a negative divided by a negative is a positive)

If you are to factor  3a² - 4ab,  factor out an a:  a(3a - 4b).

If that's not what you're asked to do, you need to give us more instructions ...

lim as x approaches ?

√(3x + 4) - 2x  =  2

To get rid of a square root, you will need to square both sides. However, before you do that, get the square root term alone on one side.

Add 2x to both sides:

√(3x + 4)  =  2x + 2

Now square both sides:

( √(3x + 4) )²  =  ( 2x + 2 )² 

       3x + 4  =  4x²  + 8x + 4

Getting all the terms to one side:

     4x²  + 5x  =  0

Factoring:

     x(4x + 5)  = 0

--->     Either  x  =  0     or     4x + 5  = 0

So, either  x = 0  or  x = -5/4.

Checking in the original problem:  x = 0  --->   √(3(0) + 4) - 2(0)  =  √(4)  =  2    Correct!

x = -5/5   --->   √(3(-5/4) + 4) - 2(-5/4)  =  √(1/4) + 10/4   1/2 + 5/2  =  6/2  =  3     Not correct!

So, the only answer is:  x = 0. 

21

2+2 is 21

it's simple

-4+-6=-10

3(-4)=-12

-10/-12=-5/-6=5/6=8.33333