Answer
a) rational
b) irrational
Work Step by Step
a) A rational number is defined as a number that can be written as $\frac{a}{b}$ where a and b are integers when b is not zero. Any terminating decimal is a rational number since it can be rewriten as a fraction where the decimal part of the number is the numerator and the denominator is the power of 10 corresponding to the rightmost digit.
$0.232\underset\uparrow4=\frac{2324}{10000}$ so it is a rational number.
$\ \ \ \ \ \ \ \ $ten-thousandths place
b) Use a calculator to find $\sqrt {46}$. The result is 6.7823299831...
This is a non-terminating decimal. Since you cannot determine the place value of the rightmost digit, a non-terminating decimal cannot be written as $\frac{a}{b}$ where a and b are integers when b is not zero. This means $\sqrt {46}$ is not a rational number, so it is an irrational number.
