# Find three ways of forming the number 100 using 3,3,5,7

As it says in the title.

Rules:

1. anything not explicitly allowed is forbidden.
2. use 3, 3, 5, 7 and multiplication, division, sum, difference, unary minus, and exponentiation in any order. Parentheses are allowed for grouping.
3. factorials, square roots, concatenation and decimal points are forbidden.
4. lateral thinking is forbidden.
5. must give three distinct expressions. No partial answers, please.

Example:

If it were 3, 5, 5, 7 instead of 3, 3, 5, 7 then you could do

1. (3+7)x(5+5)
2. (7-3)x5x5
3. 3x5x7-5

Clarification: What is distinct? Or rather, what is not distinct:

5x(-3+7)x5      -   same as 2.
(-3)x(-5)x7-5   -   same as 3.


Attribution: conceived by myself

• Rule 4 is unconditionally illegal. – Egor Hans Mar 1 at 9:29

First Way

$$100 = 7^3 - 3^5$$

Second Way

$$100 = (3^3- 7)\times 5$$

Third Way

$$100 = (7+3)^{(5-3)}$$

• I verified via computer program that these are in fact the only 3 solutions. – dshin Mar 2 at 15:42