Question

1. circle the irrational number in the list 7.27 $\frac{5}{9}$ $sqrt{15}$ $sqrt{196}$

Explanation:

Step1: Recall definition of rational and irrational numbers

A rational number can be written as a fraction $\frac{a}{b}$ ($b
eq0$) or as a terminating or repeating decimal. An irrational number cannot be written as a fraction and has a non - repeating, non - terminating decimal expansion.

Step2: Analyze $7.\overline{27}$

$7.\overline{27}$ is a repeating decimal. Let $x = 7.\overline{27}=7 + 0.27+0.0027+\cdots$. Using the formula for the sum of an infinite geometric series $S=\frac{a}{1 - r}$ (where $a = 0.27$ and $r=0.01$), $x=7+\frac{0.27}{1 - 0.01}=7+\frac{27}{99}=\frac{7\times99 + 27}{99}=\frac{693+27}{99}=\frac{720}{99}$, so it is rational.

Step3: Analyze $\frac{5}{9}$

It is already in fraction form $\frac{a}{b}$ ($a = 5$, $b = 9$), so it is rational.

Step4: Analyze $\sqrt{15}$

The square root of a non - perfect square number is irrational. Since $15$ is not a perfect square (i.e., there is no integer $n$ such that $n^2=15$), $\sqrt{15}$ has a non - repeating, non - terminating decimal expansion, so it is irrational.

Step5: Analyze $\sqrt{196}$

$\sqrt{196}=14$ because $14\times14 = 196$. It can be written as $\frac{14}{1}$, so it is rational.

Answer:

$\sqrt{15}$