overview

Repeated addition is named multiplication.

Similarly, repeated multiplication is named exponents.

• ${2}^{3}=2\times 2\times 2=8$

• The expression $2}^{3$ 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.

repeated

"Multiplication" is "repeated addition". Multiplication is learned as repeated addition in whole numbers.

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

Later, multiplication is extended to negative numbers and fractions.

eg: $3\times (-5)$ is $3$ repeated $-5$ times.

$3\times \frac{1}{6}$ is $3$ repeated $\frac{1}{6}$ times.

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

Similarly, "*exponents*" are introduced as repeated multiplication.

${3}^{5}=3\times 3\times 3\times 3\times 3$.

In this, we learn

• What is exponent to a whole number, eg : $3}^{5$?

• What is exponent to a negative number, eg: $3}^{-5$?

• What is exponent to a fraction, eg: $9}^{\frac{1}{2}$?

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.

$5\times 5\times 5={5}^{3}=125$

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

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

$2}^{5$

$=2\times 2\times 2\times 2\times 2$

$=32$

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

negative

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 $\text{aligned:}3$ in directed whole number form. To compute $2}^{3$, in the aligned direction, starting from $1$, multiply in ratio of base $2$.
It progresses in the following order

$2,2\times 2,2\times 2\times 2$

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

Integer $-3$ is $\text{opposed:}3$ 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 $\frac{1}{2},\frac{1}{2\times 2},\frac{1}{2\times 2\times 2}$.

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

summary

"*exponents*" are repeated multiplication.

${3}^{2}=3\times 3=9$

$3$ is the base

$2$ is the power

$9$ is the exponentiation result

• ${3}^{2}=3\times 3$

• $\left(\frac{1}{3}\right)}^{2}=\frac{1}{3}\times \frac{1}{3$

• ${(-3)}^{2}=(-3)\times (-3)$

• $3}^{-2}=\frac{1}{3\times 3$

• ${9}^{\frac{1}{2}}=?$ this is introduced later.

Outline

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

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