To determine which numbers in the set [tex]\( M \)[/tex] are rational, we need to evaluate each element of the set and recognize its nature. Recall that a rational number is any number that can be expressed as the quotient [tex]\( \frac{p}{q} \)[/tex] of two integers [tex]\( p \)[/tex] and [tex]\( q \)[/tex], where [tex]\( q \neq 0 \)[/tex].
Given the set:
[tex]\[
M = \left\{0, \sqrt{11}, -0.06, 0.75, 0 . \overline{7}, 36, -49, \frac{\pi}{6}, \frac{8}{\sqrt{2}}, \sqrt{-25}\right\}
\][/tex]
We will examine each element of [tex]\( M \)[/tex] one-by-one:
1. [tex]\( 0 \)[/tex]: Rational. It can be expressed as [tex]\( \frac{0}{1} \)[/tex].
2. [tex]\( \sqrt{11} \)[/tex]: Irrational. It cannot be expressed as the quotient of two integers.
3. [tex]\( -0.06 \)[/tex]: Rational. It is equal to [tex]\( -\frac{6}{100} \)[/tex] or [tex]\( -\frac{3}{50} \)[/tex].
4. [tex]\( 0.75 \)[/tex]: Rational. It is equal to [tex]\( \frac{75}{100} \)[/tex] or [tex]\( \frac{3}{4} \)[/tex].
5. [tex]\( 0.\overline{7} \)[/tex] (or [tex]\( 0.777777\ldots \)[/tex]): Rational. It is equal to [tex]\( \frac{7}{9} \)[/tex].
6. [tex]\( 36 \)[/tex]: Rational. It can be expressed as [tex]\( \frac{36}{1} \)[/tex].
7. [tex]\( -49 \)[/tex]: Rational. It can be expressed as [tex]\( \frac{-49}{1} \)[/tex].
8. [tex]\( \frac{\pi}{6} \)[/tex]: Irrational. [tex]\( \pi \)[/tex] is an irrational number, and any non-zero multiple or fraction of [tex]\( \pi \)[/tex] is also irrational.
9. [tex]\( \frac{8}{\sqrt{2}} \)[/tex]: Rational. It simplifies to [tex]\( 8 \div \sqrt{2} \)[/tex] which equals [tex]\( 8 \times \frac{\sqrt{2}}{2} = 4\sqrt{2} \)[/tex], which is still irrational because [tex]\( \sqrt{2} \)[/tex] is irrational.
10. [tex]\( \sqrt{-25} \)[/tex]: Not a real number, it's a purely imaginary number [tex]\( 5i \)[/tex].
Now, listing the rational numbers from the evaluated set, we get:
[tex]\[
\{ 0, -0.06, 0.75, 0.7777777777777777, 36, -49 \}
\][/tex]
