Correct Answer
Options A (), B (), and E () should be selected.
Explanation
A rational number is any number that can be written as a fraction, where both the numerator and denominator are integers (whole numbers). Since negative 72 \( -72 \) can be written as fraction with a numerator of negative 72 and a denominator of 1 \( \frac{-72}{1}, \) it's a rational number.
Option B, four fifths, \( \frac{4}{5}, \) is already written as a fraction with integers in the numerator (4) and the denominator (5). A number that can be expressed as a fraction is rational.
And option E, the square root of 100, \( \sqrt{100}, \) is also rational because 100 is a perfect square: 10 times 10 equals 100, \( 10 \times 10 = 100, \) so the square root of 100 = 10. \( \sqrt{100} = 10. \) Since 10 is a whole number, the square root of its square is rational.
Unlike 100, 6, in option B, is not a perfect square. There's no whole number that, when squared, gives 6. Its decimal form is non-repeating and non-terminating, which makes the square root of 6 \( \sqrt{6} \) irrational. Option C is slightly more complex. The square root of a fraction is equivalent to the square root of the numerator divided by the square root of the denominator. In this case,the square root of 5 sixtenths equals fraction with numerator of the square root of five and denominator of the square root of 16.
$$ \sqrt{\frac{5}{16}} = \frac{\sqrt{5}}{\sqrt{16}} $$
Although 16 is a perfect square, and the square root of 16 is rational, 5 is not a perfect square and its square root is irrational. Dividing an irrational number by a rational number gives you an irrational number, and option C is therefore irrational.