maths > exponents

Representation of Exponents

    what you'll learn...


Repeated addition is named multiplication.

Similarly, repeated multiplication is named exponents.

 •  23=2×2×2=8

 •  The expression 23 is, 2 to the power of 3 or 2 power 3.

The special cases where the power is a negative number or a positive fraction are explained.


"Multiplication" is "repeated addition". Multiplication is learned as repeated addition in whole numbers.
3×5 is 3 repeated 5 times (or 3 times 5).

Later, multiplication is extended to negative numbers and fractions.
eg: 3×(-5) is 3 repeated -5 times.
3×16 is 3 repeated 16 times.

We learned meaning of multiplication by whole numbers, negative numbers, and fractions.

Similarly, "exponents" are introduced as repeated multiplication.

In this, we learn
 •  What is exponent to a whole number, eg : 35?
 •  What is exponent to a negative number, eg: 3-5?
 •  What is exponent to a fraction, eg: 912?

The word "exponent" means: a number raised to the power of another number". In plain English, exponent means "higher level or higher order".

base to the power

Exponents : Exponents are repeated multiplication.

5 is the base
3 is the exponent or the power
125 is the result of exponentiation

The word "base" means: the foundation or lower part of something.

The expression 25 is, 2 to the power of 5 or 2 power 5.

The value of 25 is "32".


In the other way, 3×3×3×3 is written as "34".


Consider 2-3. The power is a negative number. To understand exponent with negative power let us revise the 'integers' (directed numbers).

Integer 3 is aligned:3 in directed whole number form. To compute 23, in the aligned direction, starting from 1, multiply in ratio of base 2. It progresses in the following order

power 3 is positive, so multiplication is repeated. For a negative power, the inverse of multiplicatio -- "division" -- is repeated.

Integer -3 is opposed:3 in directed whole number form. To compute 2-3, in the opposed direction, starting form 1 divide in ratio of base 2. It progresses in the following order 12,12×2,12×2×2.

In the other way, 13×3×3×3 is written as "3-4"


"exponents" are repeated multiplication.
    3 is the base
    2 is the power
    9 is the exponentiation result

 •  32=3×3
 •  (13)2=13×13
 •  (-3)2=(-3)×(-3)
 •  3-2=13×3
 •  912=? this is introduced later.


The outline of material to learn "Exponents" is as follows. Note: click here for detailed outline of Exponents s

    →   Representation of Exponents

    →   Inverse of exponent : root

    →   Inverse of exponent : Logarithm

    →   Common and Natural Logarithms

    →   Exponents Arithmetics

    →   Logarithm Arithmetics

    →   Formulas

    →   Numerical Expressions

    →   PEMA / BOMA

    →   Squares and Square roots

    →   Cubes and Cube roots