Question

determine whether each number is rational or irrational.
rational irrational
(0.overline{6}) (circ) (circ)
(\frac{sqrt{36}}{3}) (circ) (circ)
(-7.77) (circ) (circ)
(sqrt{15}) (circ) (circ)

Explanation:

For \( 0.\overline{6} \)

Step1: Recall rational number definition

A rational number can be expressed as a fraction \( \frac{p}{q} \) (\( q
eq0 \)), or is a repeating/terminating decimal. \( 0.\overline{6} \) is a repeating decimal. Let \( x = 0.\overline{6} \), then \( 10x=6.\overline{6} \). Subtract: \( 10x - x = 6.\overline{6}-0.\overline{6} \), so \( 9x = 6 \), \( x=\frac{6}{9}=\frac{2}{3} \), a fraction.

Step2: Classify

Since it can be written as a fraction, \( 0.\overline{6} \) is rational.

For \( \frac{\sqrt{36}}{3} \)

Step1: Simplify the expression

First, calculate \( \sqrt{36}=6 \) (since \( 6\times6 = 36 \)). Then \( \frac{\sqrt{36}}{3}=\frac{6}{3}=2 \).

Step2: Classify

\( 2 \) is an integer, and integers are rational (can be written as \( \frac{2}{1} \)). So \( \frac{\sqrt{36}}{3} \) is rational.

For \( -7.77 \)

Step1: Recall rational number definition

\( -7.77 \) is a terminating decimal. Terminating decimals are rational because they can be written as a fraction. For example, \( -7.77=-\frac{777}{100} \).

Step2: Classify

Since it is a terminating decimal (and can be expressed as a fraction), \( -7.77 \) is rational.

For \( \sqrt{15} \)

Answer:

• \( 0.\overline{6} \): Rational (circle the "Rational" option for \( 0.\overline{6} \))
• \( \frac{\sqrt{36}}{3} \): Rational (circle the "Rational" option for \( \frac{\sqrt{36}}{3} \))
• \( -7.77 \): Rational (circle the "Rational" option for \( -7.77 \))
• \( \sqrt{15} \): Irrational (circle the "Irrational" option for \( \sqrt{15} \))