Exponents of numbers

Exponents of numbers

An exponent is a quantity that shows how many times a number is repeated in multiplication. The number in exponential form is written as "$a^n$", where "$a$" is the base and "$n$" is an exponent. Also "$a^n$" is called a power. The exponent of a number is written on the top right of the base. For example, $3 \times 3 \times 3 \times 3 \times 3$ in exponential form is written as $3^5$ and it is read as three exponent five or three power five.

Writing numbers in exponential form

A number with the same factors can be written as a product of its factors. The product of the factors can be written in exponential form as shown in the following example:

Example

(a) $7 \times 7 \times 7 \times 7 \times 7 \times 7 = 7^6$

(b) $5 \times 5 \times 5 \times 5 = 5^4$

(c) $(3/5) \times (3/5) \times (3/5) \times (3/5) = (3/5)^4$

(d) $y \times y \times y \times y \times y = y^5$

Multiplication of exponential numbers (powers)

Consider two pairs of powers, that is, $a^n$, $a^m$ and $b^n$, $b^m$, where $n$ and $m$ are integers.

(a) $a^n$ and $a^m$ are powers with the same base and different exponents.

Thus, $a^n \times a^m = a^{n+m}$

(b) $a^n$ and $b^m$ are powers with different bases and exponents.

Thus, $a^n \times b^m \neq a^{n+m}$ or $a^n \times b^m \neq b^{m+n}$.

(c) $a^n$ and $b^n$ are powers with the same exponent but different bases.

Thus, $a^n \times b^n = (a \times b)^n$.

Example 1

Simplify the following expressions, and then write the answers in exponential form:

(a) $5^8 \times 5^3$ (b) $(2/5)^3 \times (2/5)^5$

(c) $(-7)^4 \times (-7)^2$ (d) $2^3 \times 4^3$

Solution

(a) $5^8 \times 5^3 = 5^{8+3} = 5^{11}$.

Therefore, $5^8 \times 5^3 = 5^{11}$.

(b) $(2/5)^3 \times (2/5)^5 = (2/5)^{3+5} = (2/5)^8$

Therefore, $(2/5)^3 \times (2/5)^5 = (2/5)^8$

Carefully study the following equations:

(a) $(a^n)^2 = a^n \times a^n$

$= a^{n+n}$

$= a^{2n}$

$= a^{n \times 2}$.

(b) $(a^n)^3 = a^n \times a^n \times a^n$

$= a^{n+n+n}$

$= a^{3n}$

$= a^{n \times 3}$.

(c) $(-7)^4 \times (-7)^2 = (-7)^{4+2} = (-7)^6$.

Therefore, $(-7)^4 \times (-7)^2 = (-7)^6$.

(d) $2^3 \times 4^3 = (2 \times 4)^3 = 8^3$.

Therefore, $2^3 \times 4^3 = 8^3$.

By considering the results obtained from parts (a) and (b) above, we conclude that $(a^n)^m = a^{n \times m}$.

Example 2

Simplify the following expressions, and then write answers in exponential form:

(a) $(3^5)^4$ (b) $(5^2)^3$

Solution

(a) $(3^5)^4 = 3^{5 \times 4} = 3^{20}$.

Therefore, $(3^5)^4 = 3^{20}$.

(b) $(5^2)^3 = 5^{2 \times 3} = 5^6$.

Therefore, $(5^2)^3 = 5^6$.