maths > exponents

Representation of Exponents

what you'll learn...

overview

Similarly, repeated multiplication is named exponents.

•  ${2}^{3}=2×2×2=8$${2}^{3} = 2 \times 2 \times 2 = 8$

•  The expression ${2}^{3}$${2}^{3}$ is, $2$$2$ to the power of $3$$3$ or $2$$2$ power $3$$3$.

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

repeated

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

Later, multiplication is extended to negative numbers and fractions.
eg: $3×\left(-5\right)$$3 \times \left(- 5\right)$ is $3$$3$ repeated $-5$$- 5$ times.
$3×\frac{1}{6}$$3 \times \frac{1}{6}$ is $3$$3$ repeated $\frac{1}{6}$$\frac{1}{6}$ times.

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

Similarly, "exponents" are introduced as repeated multiplication.
${3}^{5}=3×3×3×3×3$${3}^{5} = 3 \times 3 \times 3 \times 3 \times 3$.

In this, we learn
•  What is exponent to a whole number, eg : ${3}^{5}$${3}^{5}$?
•  What is exponent to a negative number, eg: ${3}^{-5}$${3}^{- 5}$?
•  What is exponent to a fraction, eg: ${9}^{\frac{1}{2}}$${9}^{\frac{1}{2}}$?

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.

$5×5×5={5}^{3}=125$$5 \times 5 \times 5 = {5}^{3} = 125$
$5$$5$ is the base
$3$$3$ is the exponent or the power
$125$$125$ is the result of exponentiation

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

The expression ${2}^{5}$${2}^{5}$ is, $2$$2$ to the power of $5$$5$ or $2$$2$ power $5$$5$.

The value of ${2}^{5}$${2}^{5}$ is "$32$$32$".

${2}^{5}$${2}^{5}$
$=2×2×2×2×2$$= 2 \times 2 \times 2 \times 2 \times 2$
$=32$$= 32$

In the other way, $3×3×3×3$$3 \times 3 \times 3 \times 3$ is written as "${3}^{4}$${3}^{4}$".

negative

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

Integer $3$$3$ is \text{aligned:}3$\textrm{\left(a l i g \ne d\right\rangle} 3$ in directed whole number form. To compute ${2}^{3}$${2}^{3}$, in the aligned direction, starting from $1$$1$, multiply in ratio of base $2$$2$. It progresses in the following order
$2,2×2,2×2×2$$2 , 2 \times 2 , 2 \times 2 \times 2$

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

Integer $-3$$- 3$ is $\text{opposed:}3$$\textrm{\left(o p p o s e d\right\rangle} 3$ in directed whole number form. To compute ${2}^{-3}$${2}^{- 3}$, in the opposed direction, starting form $1$$1$ divide in ratio of base $2$$2$. It progresses in the following order $\frac{1}{2},\frac{1}{2×2},\frac{1}{2×2×2}$$\frac{1}{2} , \frac{1}{2 \times 2} , \frac{1}{2 \times 2 \times 2}$.

In the other way, $\frac{1}{3×3×3×3}$$\frac{1}{3 \times 3 \times 3 \times 3}$ is written as "${3}^{-4}$${3}^{- 4}$"

summary

"exponents" are repeated multiplication.
${3}^{2}=3×3=9$${3}^{2} = 3 \times 3 = 9$
$3$$3$ is the base
$2$$2$ is the power
$9$$9$ is the exponentiation result

•  ${3}^{2}=3×3$${3}^{2} = 3 \times 3$
•  ${\left(\frac{1}{3}\right)}^{2}=\frac{1}{3}×\frac{1}{3}$${\left(\frac{1}{3}\right)}^{2} = \frac{1}{3} \times \frac{1}{3}$
•  ${\left(-3\right)}^{2}=\left(-3\right)×\left(-3\right)$${\left(- 3\right)}^{2} = \left(- 3\right) \times \left(- 3\right)$
•  ${3}^{-2}=\frac{1}{3×3}$${3}^{- 2} = \frac{1}{3 \times 3}$
•  ${9}^{\frac{1}{2}}=?$ this is introduced later.

Outline

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