Question

identify all five numbers that are not rational numbers.
a $sqrt{16}$ b $sqrt{15}$
c $sqrt{11}$ d $sqrt{9}$
e $sqrt{7}$ f $sqrt{6}$
g $sqrt{5}$ h $sqrt{1}$

Explanation:

Step1: Recall rational number definition

A rational number is a number that can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$. For square roots, $\sqrt{n}$ is rational if $n$ is a perfect square (i.e., $n = k^2$ for some integer $k$).

Step2: Analyze $\sqrt{16}$

$\sqrt{16} = 4$, which is an integer (and thus rational, since $4=\frac{4}{1}$). So A is rational.

Step3: Analyze $\sqrt{15}$

15 is not a perfect square (since $3^2 = 9$, $4^2 = 16$), so $\sqrt{15}$ is irrational (not rational). So B is not rational.

Step4: Analyze $\sqrt{11}$

11 is not a perfect square ( $3^2 = 9$, $4^2 = 16$), so $\sqrt{11}$ is irrational. So C is not rational.

Step5: Analyze $\sqrt{9}$

$\sqrt{9}=3$, which is an integer (rational, $3=\frac{3}{1}$). So D is rational.

Step6: Analyze $\sqrt{7}$

7 is not a perfect square ( $2^2 = 4$, $3^2 = 9$), so $\sqrt{7}$ is irrational. So E is not rational.

Step7: Analyze $\sqrt{6}$

6 is not a perfect square ( $2^2 = 4$, $3^2 = 9$), so $\sqrt{6}$ is irrational. So F is not rational.

Step8: Analyze $\sqrt{5}$

5 is not a perfect square ( $2^2 = 4$, $3^2 = 9$), so $\sqrt{5}$ is irrational. So G is not rational.

Step9: Analyze $\sqrt{1}$

$\sqrt{1}=1$, which is an integer (rational, $1=\frac{1}{1}$). So H is rational.

Now, we need to identify five non - rational (irrational) numbers. From the above analysis, the non - rational ones are B ($\sqrt{15}$), C ($\sqrt{11}$), E ($\sqrt{7}$), F ($\sqrt{6}$), G ($\sqrt{5}$). Wait, but let's check again. Wait, the problem says "ALL FIVE numbers that are NOT rational numbers". Wait, maybe I missed some. Wait, let's re - check:

Wait, the options: A: $\sqrt{16}$ (rational), B: $\sqrt{15}$ (irrational), C: $\sqrt{11}$ (irrational), D: $\sqrt{9}$ (rational), E: $\sqrt{7}$ (irrational), F: $\sqrt{6}$ (irrational), G: $\sqrt{5}$ (irrational), H: $\sqrt{1}$ (rational). So the irrational (not rational) ones are B, C, E, F, G.

Answer:

B. $\sqrt{15}$, C. $\sqrt{11}$, E. $\sqrt{7}$, F. $\sqrt{6}$, G. $\sqrt{5}$