Question

Which number can be written as a fraction and is therefore Rational?

8√square root of 8

0.12345678910.....

10√the square root of 10

0.89898989....

Answer 1

The numbers that can be written as fractions and are therefore rational are 8√8 and 10√10.

Step-by-step explanation:

Rational numbers are those that can be expressed as a fraction of two integers. In the given options, 8√8 and 10√10 can be rationalized to rational numbers.

Starting with 8√8, let's simplify it. 8√8 = 8 * (8^0.5) = 8 * 2 = 16. Since 16 can be expressed as the fraction 16/1, it is a rational number.

Now, consider 10√10. Similarly, 10√10 = 10 * (10^0.5) = 10 * 3.162 = 31.62. Again, this can be expressed as the fraction 3162/100, simplifying to 1581/50. Thus, 10√10 is a rational number.

Moving on to the decimal representations, 0.12345678910... is a rational number. Although it has an infinite decimal expansion, it follows a repeating pattern (0.123456789101112...), making it a rational number.

Lastly, 0.89898989... is also rational. It is a repeating decimal with the pattern 89, making it representable as the fraction 89/99.

In conclusion, all the provided numbers can be expressed as fractions, and therefore, they are rational numbers.

Answer 2

0.89898989.... is the correct answer

Fraction form:
`(89)/(99)`

Step-by-step explanation:

0.89898989.... is the correct answer because it can be written as
`(89)/(99)` in the simplest form. and 0.89898989.... is rational because it is a repeating number.

Any rational number is a number that can be written as a fraction, a repeating decimal, a whole number, an integer, etc. As long as the number is a number with infinite digits like π.