3f^4g^3 x 8fg^4 / (6f^2g^3)^2 how do you do this? I know the answer is 2/3fg but how do you get 0.666... into that fraction form?

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  0.666... = 6/9 = 2/3

Step-by-step explanation:

The repeating decimal 0.666... can be turned to a fraction as follows.

Let x represent the value of the number. Then 10x is ...

  10x = 10(0.666...) = 6.666...

Subtracting x gives ...

  10x -x = 6.666... - 0.666... = 6.000...

  9x = 6

Dividing by 9 and simplifying, we have ...

  x = 6/9 = 2/3

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It can be worthwhile to remember that  1/9 = 0.111... repeating.

Then 3/9 = 1/3 = 3(0.111...) = 0.333... repeating

and 6/9 = 2/3 = 6(0.111...) = 0.666... repeating

Most calculators will round the last digit of 2/3 so that it is a 7. So, you may see ...

  2 ÷ 3 = 0.6666667 or 0.66666666667

or something similar, depending on your calculator. It is up to you to recognize this as probably a representation of 2/3. Some calculators will display the number as a fraction, so you don't have to guess.

Additional comment

A similar method is used to convert repeating decimals of other lengths to fractions. In general you multiply by 10^n, where n is the number of repeating digits. Then proceed as above: subtract x, divide by the coefficient of x. You will find you are always dividing by a number that has as many 9's as there are repeating digits.

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step-by-step explanation: