1. ## Cute stuff

1^1 + 3^2 + 5^3 = 135
1^1 + 7^2 + 5^3 = 175

1^3 + 3^3 + 6^3 = 244
2^3 + 4^3 + 4^3 = 136

2. Nice. You should check Posamentier's Math Charmers

3. Merci, WB.

1^3 + 5^3 + 3^3 = 153

3^3 + 7^3 + 0^3 = 370

3^3 + 7^3 + 1^3 = 371

4^3 + 0^3 + 7^3 = 407

1^4 + 6^4 + 3^4 + 4^4 = 1634

8^4 + 2^4 + 0^4 + 8^4 = 8208

9^4 + 4^4 + 7^4 + 4^4 = 9474

4. Nice!

5. . . $\begin{array}{ccc}
4^5 + 1^5 + 5^5 + 0^5 &=& 4150 \\ \\
5^5 + 4^5 + 7^5 + 4^5 + 8^5 &=& 54748 \\ \\
9^5+2^5+7^5+2^5+7^5 &=& 92727 \end{array}$

6. . . $\begin{array}{ccc}
1^7 + 7^7 + 4^7 + 1^7 +7^7 + 2^7 + 5^7 \;\;=\;\; 1,\!741,\!725 \\ \\

2^8 + 4^8 + 6^8 + 7^8 + 8^8 + 0^8 + 5^8 + 0^8 \;\;=\;\; 24,\!678,\!050 \\ \\

4^{10}+6^{10}+7^{10}+9^{10}+3^{10}+ 0^{10} + 7^{10} + 7^{10} + 7^{10} + 4^{10} \;\;=\;\; 4,\!679,\!307,\!774 \\ \\

4! + 0! + 5! + 8! + 5! \;\;=\;\; 40,\!585 \\ \\

\dfrac{3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2}{1^2+2^2+3^2+4^2+5^2+6^2+7^2} \;\;=\;\;2
\end{array}$

7. $99,999,999 = \displaystyle{\left( 9 +\frac{9}{9}\right)^{9 -\frac{9}{9}} - \frac{9}{9}}$

8. My own, fresh off the press: 4^0 - 3^3 + 7^9 - 5^6 = 4 0 3 3 7 9 5 6

9. $666+6+6+6 = (6+6/6)^{(6+6+6)/6} + (6-6/6)^{(6+6+6)/6} + 6^{(6+6+6)/6}$

10. Originally Posted by bandedkrait
$666+6+6+6 = (6+6/6)^{(6+6+6)/6} + (6-6/6)^{(6+6+6)/6} + 6^{(6+6+6)/6}$
BEASTily interesting!!

11. . . . $\begin{array}{c}
\text{Some Beastly Identities} \\ \\ \\
1 + 2 + 3 + 4 + 567 + 89 \\ \\
123 + 456 + 78 + 9 \\ \\
9 + 87 + 6 + 543 + 21 \\ \\
1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 5^3 + 4^3 + 3^3 + 2^3 + 1^3 \\ \\
2^2+3^2+5^2+7^2+11^2+13^2+17^2\\
^{\text{(First seven primes)}}
\end{array}$

12. Originally Posted by Wilmer
BEASTily interesting!!
Haha . Credit goes to Prof. John Barrow. It's from his popular science book, "The Constants of Nature."

13. ## Here's my small contribution

16^3 + 50^3 + 33^3 = 165,033.

14. 10^2*2 + 10^1*5 + 10^0*4 = 254

oh no, thats cheating isn't it.

15. Originally Posted by alan2
10^2*2 + 10^1*5 + 10^0*4 = 254

oh no, thats cheating isn't it.
I think this is a consequence of the definition of the decimal system.