Division of exponential numbers

Division of exponential numbers

Consider exponential numbers with the same base, for example $a^n$ and $a^m$. You can simply divide the numbers with the same base as follows:

$a^5 \div a^3 = \frac{a^5}{a^3} = \frac{a \times a \times a \times a \times a}{a \times a \times a} = a^2$

Thus, $a^5 \div a^3 = a^{5-3} = a^2$.

This example shows that you have to subtract the exponents when dividing numbers with the same base.

Therefore, $a^n \div a^m = a^{n-m}$.

Exponential numbers with the same exponents and different bases such as $a^n$ and $b^n$ can be divided as follows:

$a^n \div b^n = \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$

Example 1

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

(a) $7^9 \div 7^5$ (b) $\left(\frac{2}{7}\right)^7 \div \left(\frac{2}{7}\right)^3$

(c) $5^4 \div 3^4$

Solution

(a) $7^9 \div 7^5 = 7^{9-5} = 7^4$

Therefore, $7^9 \div 7^5 = 7^4$

(b) $\left(\frac{2}{7}\right)^7 \div \left(\frac{2}{7}\right)^3 = \left(\frac{2}{7}\right)^{7-3} = \left(\frac{2}{7}\right)^4$

Therefore, $\left(\frac{2}{7}\right)^7 \div \left(\frac{2}{7}\right)^3 = \left(\frac{2}{7}\right)^4$

(c) $5^4 \div 3^4 = \frac{5^4}{3^4} = \left(\frac{5}{3}\right)^4$

Therefore, $5^4 \div 3^4 = \left(\frac{5}{3}\right)^4$

Zero exponent

When dividing numbers with the same base and exponent, you will obtain a base with a zero exponent. The following example shows the value of a base with a zero exponent.

Example 2

$a^3 \div a^3 = \frac{a^3}{a^3} = \frac{a \times a \times a}{a \times a \times a} = 1$

Remember that $a^3 \div a^3 = a^{3-3} = a^0$

Therefore, $a^3 \div a^3 = 1$. This shows that $a^0 = 1$.

Simplification of exponential expressions

Example 3

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

(a) $2^3 \times 3^4 \times \frac{4}{3} \times 3^2$

(b) $\frac{8a^5b^3 + 4a^3b^4}{2a^3b^3}$

Solution

(a) Find the prime factors of 32 and 4; then write their products in exponential form:

$32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5$; $4 = 2 \times 2 = 2^2$

$2^3 \times 3^4 \times \frac{4}{3} \times 3^2 = 2^3 \times 3^4 \times \frac{2^2}{3} \times 3^2$

$= 2^{3+2} \times \frac{3^4}{3} \times 3^2 = 2^5 \times \frac{3^4}{3} \times 3^2$

$= \frac{2^5}{2^5} \times \frac{3^4}{3} = 2^{5-5} \times 3^{4-1} = 2^0 \times 3^3 = 1 \times 33 = 33$

Therefore, $2^3 \times 3^4 \times \frac{4}{3} \times 3^2 = 33$

(b) This example shows how to simplify exponential algebraic expressions.

$\frac{8a^5b^3 + 4a^3b^4}{2a^3b^3} = \frac{8a^5b^3}{2a^3b^3} + \frac{4a^3b^4}{2a^3b^3}$

$= \frac{8}{2}(a^5 \div a^3 \times b^3 \div b^3) + \frac{4}{2}(a^3 \div a^3 \times b^4 \div b^3)$

$= 4(a^{5-3} \times b^{3-3}) + 2(a^{3-3} \times b^{4-3})$

$= 4(a^2 \times b^0) + 2(a^0 \times b^1)$

$= 4(a^2 \times 1) + 2(1 \times b)$

$= 4a^2 + 2b$

$= 2(2a^2 + b)$

Alternative solution 1

$\frac{8a^5b^3 + 4a^3b^4}{2a^3b^3} = \frac{8a^5b^3}{2a^3b^3} + \frac{4a^3b^4}{2a^3b^3}$

$= 4a^2(1) + 2(1)b = 4a^2 + 2b = 2(2a^2 + b)$

Therefore, $\frac{8a^5b^3 + 4a^3b^4}{2a^3b^3} = 2(2a^2 + b)$

Alternative solution 2

$\frac{8a^5b^3 + 4a^3b^4}{2a^3b^3} = \frac{4a^3b^3(2a^2 + b)}{2a^3b^3} = 2(2a^2 + b)$

Therefore, $\frac{8a^5b^3 + 4a^3b^4}{2a^3b^3} = 2(2a^2 + b)$