Exponents : Representation of Exponents

maths > exponents

Representation of Exponents

what you'll learn...

overview

Repeated addition is named multiplication.

Similarly, repeated multiplication is named exponents.

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

• 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.
$3 \times 5$ $3xx5$ is $3$ $3$ repeated $5$ $5$ times (or $3$ $3$ times $5$ $5$ ).

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

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

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

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: $9^{\frac{1}{2}}$ $9^(1/2)$ ?

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.

$5 \times 5 \times 5 = 5^{3} = 125$ $5xx5xx5 = 5^3 = 125$
$5$ $5$ 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 " $32$ $32$ ".

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

In the other way, $3 \times 3 \times 3 \times 3$ $3xx3xx3xx3$ 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 $aligned: 3$ $text(aligned:)3$ 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
$2, 2 \times 2, 2 \times 2 \times 2$ $2, 2xx2, 2xx2xx2$

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

Integer $- 3$ $-3$ is $opposed: 3$ $text(opposed:)3$ 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 $\frac{1}{2}, \frac{1}{2 \times 2}, \frac{1}{2 \times 2 \times 2}$ $1/2, 1/(2xx2), 1/(2xx2xx2)$ .

In the other way, $\frac{1}{3 \times 3 \times 3 \times 3}$ $1/(3xx3xx3xx3)$ is written as " $3^{- 4}$ $3^(-4)$ "

summary

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

• $3^{2} = 3 \times 3$ $3^2 = 3 xx 3$
• ${(\frac{1}{3})}^{2} = \frac{1}{3} \times \frac{1}{3}$ $(1/3)^2 = 1/3 xx 1/3$
• ${(- 3)}^{2} = (- 3) \times (- 3)$ $(-3)^2 = (-3) xx (-3)$
• $3^{- 2} = \frac{1}{3 \times 3}$ $3^(-2) = 1/(3 xx 3)$
• $9^{\frac{1}{2}} = ?$ $9^(1/2) = ?$ 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