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9 ( 4 ( 2n + r ) - 11r ) + 3 ( r - n )

Work from the innermost parentheses first: Solve 4 ( 2n + r ) by using the distributive property:

    4 ( 2n + r  ) =  4 x 2n + 4 x r   =   8n + 4r

This gives us:   9 ( 8n + 4r - 11r ) + 3 ( r - n )

Still working inside the first set of parentheses, combine   4r   with   - 11r   to give   - 7r.

We now have:   9 ( 8n - 7r ) + 3 ( r - n ).

Now use the distributive property on both sets of parentheses:

    9 ( 8n - 7r )   =  9 x 8n  -  9 x 7r   =   72n - 63r                         3 ( r - n )   =  3 x r  -  3 x n   =   3r - 3n

This becomes:   72n - 63r + 3r - 3n

Which is:   69n - 60r 

You need to solve for one variable first, b/c you cant combine the two variables in any way.

My calculator tells me that 1 + 1 = 2

Since your problem is x^4 = 81, what you have to undeo is the "to the fourth power" part.

The inverse (opposite operation) of "to the fourth power" is "to the fourth root".

   (x^4) ^ ¼  = 81 ^ ¼ 

   81 = 3^4    so (3 ^ 4) ^ ¼  =   3 ^ 1   = 3

By hand: (3 ^ 4) ^ ¼  =  3 ^ (4 x ¼) = 3 ^ 1  = 3   (When you take a power to a power, multiply the exponents.)

Or, do it by calculator. One way of finding the fourth root is to use ^ (1/4)

You cannot!

Say that four people share 12 apples, how many apples does each get? three, right?

What about three people? 12/3 = 4. Four apples each.

What about two people? Six apples.

If only one person "shares" 12 apples, he get all the 12 apples for himself.

What, then, if zero people "share" 12 apples? Does zero people get 12 apples? That doesn't make any sense! Does zero people get zero apples? Then, what happens with the 12 apples? All of this is nonsense, one simply cannot divide by zero.

I've linked a wikipedia article about the matter:

http://en.wikipedia.org/wiki/Division_by_zero

Division by zero is undefined because it produces no single answer.

To check division, multiply. For example    14 ÷ 7 = 2   because   7 X 2 = 14. 

If you have any non-zero number, such as 6, and divide it by zero, there is no possible answer.

        6 ÷ 0 = ?      To solve this, you need an answer for which ? X 0 = 6. This is impossible. No number works!

If you take 0 and divide it by 0, you get a different problem, because every number works!

       0 ÷ 0 = ?       Here, you can use any number for an answer. 

                            0 ÷ 0 = 27   because 0 X 27 = 0

                            0 ÷ 0 = -4   because 0 X -4 = 0    Similarly, every number works.

In the first case, you get no answer, in the second case you get no single answer.

3to4=81 

As a decimal approximation, √896 ≈ 29.93.

If you want your answer in simplest radical form, find a perfect square (4, 9, 16, 25, 36, 49, 64, 81, ...) that divides into 896 evenly (no remainder):

896 ÷ 4 = 224

224 ÷ 4 = 56

56 ÷ 4 = 14  

(14 cannot be divided evenly by any perfect square, so stop) 

There have been three successful divisions by 4,

So, 896 = 4 x 4 x 4 x 14    

--->   √896 = √4 x V4 x V4 x √14    =    2 x 2 x 2 x √14    =    8 x √14    =    8√14

1) Using the quadratic formula: 2 ± 2i√3

2) 0° Use "their", not "there".

'MATHS" -- Do you have the SATs in the British Empire?

Thank you so much! 

w^6 · x^5 / ( w^2 · x^3 )    =    3^6 · 4^5 / ( 3^2  ·  4^3 ) = 1296

This problem can be reduced: the w^6 / w^2 becomes w^4

                                       and the x^5 / x^3 becomes x^2

so the problem reduces to     w^4  · x^2    =   3^4 · 4^2    =    1296

Hit (2nd) tan  --- which will give you tan−¹ --- because you want  tan−¹(-1)

Sun = 1.9 x 10^30 = 19 x 10^29

-----------------------

Earth = 5.9 x 10^24 ---> if we want to find how many earths that it would take to make a sun, we need to divide the size of the sun by the size of the earth:    

(19 x 10^29) ÷ (5.9 x 10^24 ) = (19 ÷ 5.9) x (10^29 ÷ 10^24) ≈ 3.2 x 10^5

(Divide 19 by 5.9 to get 3; divide 10^29 by 10^24 to get 10^5 (subtract exponents).)

------------------------

Venus = 4.9 x 10^24 ---> Dividing the size of the sun by the size of Venus:

(19 x 10^29) ÷ (4.9 x 10^24 ) = (19 ÷ 5.9) x (10^29 ÷ 10^24) ≈ 3.9 x 10^5

I don't know do you know?

√-25 is not real because there is no real number, which when multiplied times itself, results in -25

5 x 5 = 25; -5 x -5 = 25 (these are the only two reasonable possibilities)

√-25 = √(-1 * 25) = √-1 * √25 =  √-1 * 5

The symbol 'i' is used to represent √-1, so the answer is usually written as 5i., where i is an imaginary number, not a real number.

Being a straight multiplication problem, my calculator gives 75014.4

The domain is defined to be the possible x-values, so if x<-1, the domain consists of those numbers that are less than -1.

The range is defined to be the possible y-values, so if y = 2x + 8 and x < -1,

replacing x with -1, we have y = 2(-1) + 8 = 6. Thus if x < -1, y < 6.

Domain: -∞ < x < -1

Range:   -∞ < y < 6

Nice job on that second one Melody !!!

There may be some factoring trick for the first, but I would have probably taken Melody's approach and used the Factor Theorem, realizing that there can be no positive real roots because all the linear terms in the original question would be > 0. And multiplying these together and adding a "+15" to them could never give us "0."

The expansion of the original question is x^4 + 16X^3 + 86x^2 +176x + 120, and the  factorization of this  is, indeed, (x+2) (x+6) (x^2 + 8x + 10)

Using "p/q,"  the first root ... (-2)... could be found pretty easily...and after some synthetic division, the resulting polynomial would be  ...   x^3 + 14x^2 + 58x + 60......and the next rational roots that would come to mind to "test" would be -3, -4, -5 and -6. And, as Melody found, -6 is the correct one.

Then, we would use synthetic division again to reduce the polynomial to x^2 + 8x + 10.... and this polynomial has no rational zeroes. (But it does have two "real" roots!!)

Again...good job, Melody !!!   Gin and tonics really DO help one's math skills.....(it's an "inside" joke !!!)

I propose a "toast".........

Move the decimal point until it is after the first non-zero digit. 

This will require that you move it 5 places to the left; each time you move it one place to the left, you are multiplying by 10. In this case, you are multiplying by 10^5.

490000.0001 = 4.900000001 x 10^5

Keep each of the zero digits that are between the 9 and the 1.

Density = mass/volume

With symbols:

d=m/V  or

ρ = m/v

ρ is pronunced "rho"

Generally, it means the amount of mass divided by the amount of volume.

For example, the units might be grams per cubic centimeters.

I would argue that if you want the equation to have a positive root, then x must be positive.

But if x is positive, then both x^15 is positive and x^5 is positive.

Adding two positive number to the number 2, your answer must be positive; thus, it can't be zero.

(proof by contradiction)

10^-2= 1/10^2 = 1/100

If you have a whole number with a negative exponent, place the whole number in the denominator and make the exponent positive. 

Write the answer as 1 over the whole number with a positive exponent.