Roots - Revision Notes

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In these revision notes for Roots, we cover the following key points:

  • What are roots?
  • What is the geometrical meaning of square and cube roots?
  • What is the closest operation to roots?
  • Why roots are called fractional powers?
  • What is the absolute value of a number?
  • What are the properties of roots? Where do they derive from?
  • How to calculate the square root of a number manually?

Roots Revision Notes

Finding the root of a number represents the inverse operation of a raise in power.

By definition, the nth root of a number is a new number that when multiplied n times by itself gives the original number.

The symbol we use to represent roots is (√). The nth root of a number x is symbolically written as x. As a special case of roots, we can mention the 2nd root of a number, known as the "square root". In the case of square roots, the index n = 2 is not written but implied, just like the 'plus' symbol before positive numbers.

A positive number has two square roots: one positive and another negative. On the other hand, the square root of negative numbers does not exist in the set of real numbers, as we cannot find any negative number which when raised at the second power gives a positive value. This is true not only for square root but for all even roots, as we can pair them two by two.

It is called 'square root' because when the area of a square is known, we calculate the side length by taking the square root of the value representing the area.

Another root that has a geometry-based meaning is the "cube root" of a number. It represents a new number which when raised to the third power gives the original number. We write the index 3 at the upper left part of the root symbol to highlight the fact that we are dealing with the third root (or cube root) of a number.

It is called 'cube root' because when the volume of a cube is known, we calculate the side length by taking the cube root of the value representing the volume. Cubic root of a number has the following features:

The cubic root of a number is a single value, not two numbers with opposite signs as in square root.

Unlike in square root, the cubic root of negative numbers does exist and it is negative.

Roots are nothing more but simply fractional powers. This means the expressions

x and x1/n

are equivalent (they show the same thing).

Roots have the following properties

Property 1: For any real non-negative numbers a and b,

√a ∙ √b = √(a ∙ b)

Property 2: For any real non-negative number a and positive real number b,

√a/√b = √(a/b)

Property 3: For any real numbers a and n,

(√a)n = √(an )

Property 4: For any real numbers a, b and for any positive integer n ≥ 2,

ab = ab/n

Property 5: For any number a and any integer n ≥ 2,

an = a if n is odd

and

an = |a| if n is even

The symbols (||) indicate the absolute value of a number that is its distance from the origin regardless the direction in the number axis.

Property 6: For any real a and b and for any positive integer n,

a ∙ b = √(an ∙ b)

Property 7: For any real number a and b for any integers m and n,

a = am

Property 8: For any real number a for any integers m and n,

a= a

There is a method that is apparently similar to division (but it is not instead) that is used to manually calculate the square root of a number.

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