<s>[INST] How can I order these numbers without a calculator? 
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*Classify the following numbers as rational or irrational. Then place them in order on a number line:


$$\pi^2, -\pi^3, 10, 31/13, \sqrt{13}, 2018/2019, -17, 41000$$
I know $\pi$ is irrational so $\pi^2$ and $-\pi^3$ are irrational. 10, 31/30, 2018/2019, -17, 41000 are all rational because then can be written in the form $\frac{a}{b}, a,b \in \mathbb{Z}, b\neq 0$
I'm not sure how I can look at $\sqrt{13}$ and determine if it's rational or irrational. 
For ordering them I am approximating $\pi$ as 3, so 
$\pi^2 \approx 9$ and 
$-\pi^3 \approx -27$. 
$31/13 \approx 30/10 = 3$
$\sqrt{9}<\sqrt{13}<\sqrt{16}$ so $3<\sqrt{13}<4$
$2018/2019 \approx 1$
If I were to order these, I would say:
$-\pi^3, -17, 2018/2019, 31/13, \sqrt{13}, \pi^2, 41000$


*Put these numbers in order from least to greatest (no calculator): 


$$10^8, 5^{12}, 2^{24}$$
All I can think of is that $2^{10} \approx 10^3$ so 
$2^{24} = (2^{10})^{2}* 2^{4} \approx (10^3)^2 *2^4 = 10^6 *16 <10^{8}$
So $2^{24} < 10^8$
Not sure how $5^{12}$ fits in.
 [/INST] $\sqrt{13}$ is irrational, by user.
The order of your first question should be $-\pi^3,-17,2018/2019,31/13,\pi^2,10,41000$.
$-\pi^3<-3^3 = -27<-17$, so we can compare two negative numbers.
We now compare $\pi^2$ and $10$, others are easy to compare. Note that $\pi = 3.1415926\dots$ and so $\pi<3.15$. Now $3.15^2 = 9.9225<10$, so $\pi^2<3.15^2<10$.
For your second question, just divide them respectively.
$\frac{10^8}{5^{12}} = \frac{2^85^8}{5^{12}} = \frac{2^8}{5^4} = \frac{4^4}{5^4}<1$, and $\frac{10^8}{2^{24}} = \frac{2^85^8}{2^{24}} = \frac{5^8}{2^{16}} = \frac{5^8}{4^8}>1$. Thus $2^{24}<10^8<5^{12}$.
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