# Pythagorean Power: Exponential Excellence

## This math trick is like an extended version of the Pythagorean Theorem!

## The examples below show why!

(x^2 + x^2)^½ = (2x^2)^½ = x * (2)^½ or x * 1.4142...

(That irrational number is the square root of 2)

The **caret** (^) is the symbol of exponentiation. Raising a number to the power of ½ gives you the square root.

## What number is **x** equal to in all these examples? Any number you want it to be!

(x^2 + x^2 + x^2)^½ = (3x^2)^½ = x * (3)^½ or x * 1.7320...

(That irrational number is the square root of 3)

(x^2 + x^2 + x^2 + x^2)^½ = (4x^2)^½ = x * (4)^½ or 2x

(4 has another integer for a square root, which is 2)

(x^2 + x^2 + x^2 + x^2 + x^2)^½ = (5x^2)^½ = x * (5)^½ or x * 2.2360...

(That irrational number is the square root of 5)

### If *y* has a rational square root, then *x* will be multiplied by a rational number!

Remember that multiplication is actually glorified addition. In the function on top of this page, the variable **y** tells you how many times you should add **x** (or rather its square) to itself. In other words, if **y** = **7**, then **x** is multiplied by **7** so you get **x** + **x** + **x** + **x** + **x** + **x** + **x** = **7****x**. Exponentiation is glorified multiplication. (**x** * **x** * **x** = *x*^3) In algebra, exponentiation is usually done before multiplication, which is usually done before addition; higher calculation levels are done before lower ones. However, **parentheses** tell us to do the math between them first before doing any math on the outside!

**Addition & subtraction** are on the same level; the lowest level of calculation. The next higher level is **multiplication & division**; then **exponentiation, radicals & logarithms**, which is the highest. This is called the "**P**lease **E**xcuse **M**y **D**ear **A**unt **S**ally" Rule.

### Although this trick works with all numbers, it's much more interesting with integers!

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© Derek Cumberbatch