Alternative University

Analytic Geometry

Squares and Cubes


The exponent o of a variable specifies multiplication of the variable into itself: the exponent specifies how many instances of the variable are involved in the multiplication. Here are examples for exponents equal to 2 and 3:

x² = x · x
x³ = x · x · x

When the exponent is 2, the variable is said to be squared. When the exponent is 3, it is cubed.

Setting x = 3 for these examples:

  =  3 · 3  =  9
  =  3 · 3 · 3  =  27

The inverse of a square and cube is the square root and cube root respectively, denoted with a radical sign ( and ) or fraction exponent:

Setting x = 3:

The square root symbol (√ ) is called a radical, and the expression within the radical is called the radicand. In this example, the radicand is 9.

A square root that produces a positive root (as shown here) is called the principal square root.  A square root could be positive or negative. In this example, the square root of 9 could be 3 or −3, referred to as ±3. However, the principal square root is always positive, in this case 3.


A unit square is an equal-sided rectangle, with each side equal to 1. For example, a square meter is a square with each side one meter in length:

Figure 1: Unit square, 1 meter by 1 meter.

We can consider a number to be that many unit squares. For example, the number 3 is three unit squares:

Figure 2: Three unit squares (each 1 meter by 1 meter).

A square, of a number of “somethings”, is a rectangle of those somethings, with that many somethings on each side (and additional somethings filling in the rest of the square if there is more than one something on each side).

Let's square 3 (which would be 3², also referred to as “3 squared”). If we have three unit squares, the square of those three unit squares is a larger square with 3 unit squares on each side, plus additional unit squares filling in the rest of the larger square (for a total of 9 squares):

Figure 3: Three squared.

Five squared has 5 units on each side, with 25 unit squares total:

Figure 4: Five squared.

Cubing is the same, using unit cubes instead of unit squares, with that many cubes wide, high and deep, and the remaining space within the new cube filled in with unit cubes. For example, 5 cubed is:

Figure 5: Five cubed.


The hypotenuse is the long side of a right triangle, also called a diagonal, or radius (because it is equal in length to the radius of a circle that has the other two sides of the triangle as horizontal and vertical offsets to a point on the circle).

Following is a right triangle with short sides labelled a and b, and the hypotenuse labelled c:

Figure 6: Right triangle.

This is the well-known 3,4,5 triangle o (a right triangle with side lengths a = 3, b = 4 and c = 5).

According to the Pythagorean theorem o, the following formula holds for any right triangle:

c² = a² + b²

where c is the hypotenuse and a and b are the short sides of the right triangle (as illustrated in the example of Figure 6 above).

We can show this formula is valid, by building the squares of the short sides, then moving those squares to the hypotenuse.

Continuing with an example 3,4,5 triangle, first build the squares of the shorter sides:

Figure 7: Shorter sides squared.

Then move 4² over to the hypotenuse:

Figure 8:  4² moved to the hypotenuse.

And move 3² over to the hypotenuse:

Figure 9:  Part of 3² moved to the hypotenuse.

Figure 10:  The rest of 3² moved to hypotenuse.

showing that 5² = 3² + 4².

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2021–Sep–22  20:09  UTC