\require{AMSmath} \require{eqn-number}

Square and Square Root

(last update:   16 Feb 2013)

The following is a bit of "the long way around the barn" but it may help see how to work your way through something mathematically, and not just square and square root functions.

Square and Square Root

First up is the square. This is simply a value multiplied by itself. For now the symbol "\cdot" is used to denote multiplication. When this isn't true, which sometimes happens in mathematics, I'll let you know. One needs to keep an open mind, as far as symbology in mathematics is concerned.

\label{eqn:TheSquare} y = x \cdot x
\label{eqn:TheSquareFunction} y = x^2

The value y is said to be the square of x. The value x is said to be the square root of the value y.

There is a special symbol used to indicate the square root operation,

\label{eqn:TheSquareRootFunction} x = \sqrt{y}

Equation\ref{eqn:TheSquareRootFunction} is saying "x is the number that, when multiplied by itself, equals y.

Other squares can be defined,

\label{eqn:TheSquareRootFunction2} y_2 = w^2
\label{eqn:TheSquareRootFunction3} y_3 = z^2

the subscripts on the y terms are used to indicated that the y terms are not all the same value.

The squares can be multiplied together,

\label{eqn:ThreeSquares} y \cdot y_2 \cdot y_3 = x^2 \cdot w^2 \cdot z^2

which can be expanded to,

\label{eqn:ThreeSquaresExp} y \cdot y_2 \cdot y_3 = (x \cdot x) (w \cdot w) (z \cdot z)

using the associative property for multiplication, we can re-group things,

\label{eqn:ThreeSquaresAssoc1} y \cdot y_2 \cdot y_3 = (x \cdot w \cdot z) (x \cdot w \cdot z)

which can be re-written as,

\label{eqn:ThreeSquaresAsSingle} y \cdot y_2 \cdot y_3 = (x \cdot w \cdot z)^2

Note that the left side can be written as,

\label{eqn:LeftSideOfThreeElementSquare} y_4 = y \cdot y_2 \cdot y_3


\label{eqn:ThreeElementSquare} y_4 = (x \cdot w \cdot z)^2

Given the definition of equation\ref{eqn:TheSquareRootFunction}, equation\ref{eqn:ThreeSquaresAsSingle} can be written as,

\label{eqn:ThreeElementRoot} x \cdot w \cdot z = \sqrt{y_4}

Expanding the right side,

\label{eqn:ThreeElementRootExpanded} x \cdot w \cdot z = \sqrt{y \cdot y_2 \cdot y_3}

Because of equations\ref{eqn:TheSquare},\ref{eqn:TheSquareRootFunction2}, and \ref{eqn:TheSquareRootFunction3} we can write,

\label{eqn:SquareRootDecomposition} \sqrt{y} \cdot \sqrt{y_2} \cdot \sqrt{y_3} = \sqrt{y \cdot y_2 \cdot y_3}


\label{eqn:SquareRootDecompositionSimple} \sqrt{y} \cdot \sqrt{y_2} \cdot \sqrt{y_3} = \sqrt{y_4}

This means that the term inside a square root operation can be decomposed to a product of square roots.