Is 0.9876.... a rational or irational number?

Answer 1

The number \( 0.9876\ldots \) is a repeating decimal, which means that the digit sequence "9876" repeats indefinitely. Any repeating decimal can be expressed as a fraction, which makes it a rational number.

To show that it's rational, you can denote the number as \( x = 0.987698769876... \). By manipulating this expression, you can convert it to a fraction. However, since \( x \) repeats, we can also recognize directly that it can be expressed as a fraction using standard techniques for converting repeating decimals to fractions.

Thus, \( 0.9876\ldots \) is a rational number.

Answer 2

what is the square root of 78?

Answer 3

The square root of 78 is an irrational number. It can be approximated as follows:

\[ \sqrt{78} \approx 8.83176 \]

If you want a simplified radical form, you can express \( \sqrt{78} \) as:

\[ \sqrt{78} = \sqrt{39 \times 2} = \sqrt{39} \sqrt{2} \]

However, since neither 39 nor 2 is a perfect square, \( \sqrt{78} \) cannot be simplified further in terms of radicals. Therefore, the best way to represent it is as \( \sqrt{78} \) or the decimal approximation \( \approx 8.83176 \).