Question

Convert 0.72¯¯¯¯¯
to a rational number in simplest form.(1 point

Answer 1

Let \( x = 0.72\overline{72} \).

To eliminate the repeating decimal, we can multiply \( x \) by 100 (since there are two digits in the repeating part) to shift the decimal point:

\[
100x = 72.72\overline{72}
\]

Next, we'll subtract the original \( x \) from this equation:

\[
100x - x = 72.72\overline{72} - 0.72\overline{72}
\]

This simplifies to:

\[
99x = 72
\]

Now, we solve for \( x \):

\[
x = \frac{72}{99}
\]

Next, we simplify the fraction \( \frac{72}{99} \). We can find the greatest common divisor (GCD) of 72 and 99:

- The prime factorization of 72 is \( 2^3 \times 3^2 \).
- The prime factorization of 99 is \( 3^2 \times 11 \).

The GCD is \( 9 \) (since \( 3^2 \) is the highest power of 3 common to both).

Now, we divide the numerator and the denominator by their GCD:

\[
\frac{72 \div 9}{99 \div 9} = \frac{8}{11}
\]

Thus, \( 0.72\overline{72} \) as a rational number in simplest form is:

\[
\boxed{\frac{8}{11}}
\]