Question

this unit test check and show all your work, especially for the written problems. good luck! question 1 which number is irrational? √7 0.75 3 0.121212…

Explanation:

Step1: Recall irrational number definition

An irrational number is a number that cannot be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$, and it has a non - repeating, non - terminating decimal expansion. Also, square roots of non - perfect squares are irrational.

Step2: Analyze $\sqrt{7}$

The number 7 is not a perfect square (since $2^2 = 4$ and $3^2=9$), so $\sqrt{7}$ is an irrational number.

Step3: Analyze 0.75

0.75 can be written as $\frac{3}{4}$, where $p = 3$ and $q = 4$ are integers with $q
eq0$. So, 0.75 is a rational number.

Step4: Analyze 3

3 can be written as $\frac{3}{1}$, where $p = 3$ and $q = 1$ are integers with $q
eq0$. So, 3 is a rational number.

Step5: Analyze 0.121212...

0.121212... is a repeating decimal. A repeating decimal can be expressed as a fraction. Let $x=0.121212\cdots$. Then $100x = 12.121212\cdots$. Subtract $x$ from $100x$: $100x - x=12.1212\cdots - 0.1212\cdots$, which gives $99x = 12$, so $x=\frac{12}{99}=\frac{4}{33}$. So, 0.121212... is a rational number.

Answer:

$\sqrt{7}$