## How to Divide Exponents

So, in layman terms, exponent of a number is basically referred to as the power of that number. So, for example, we have often seen this phrase in mathematics “$$2$$ to the power $$4$$”. So, this is nothing but $$2^4$$. This generally means that $$2$$ is multiplied by itself $$4$$ times. So, the general expression could be written as $$2 \times 2 \times 2 \times 2$$.
Exponents are basically used to reduce the length of an algebraic expression. We know how tedious would it be if we had to write “$$2$$ to the power $$10$$” in the general form without the exponent. $$2^10$$ is a very simple and less tedious way to write this.
Now, let’s consider the exponent case of $$2^4$$. In this case, $$2$$ is referred to as the "base," while $$4$$ is referred to as the "exponent" or "power." In the term $$x^n$$:

• $$n$$ is the exponent or power
• $$x$$ is called the base

### Properties of Exponents

When dealing with exponents, it is necessary to apply the properties of exponents or the rules of exponents in order to solve the problem. These characteristics are also referred to as major exponents rules, which must be observed when solving exponent problems. The next section discusses the properties of exponents.

• The Product Law states that $$a^m \times a^n \ = \ a^{m \ + \ n}$$
• The Law of Quotients states that $$\frac{a^m}{a^n} \ = \ a^{m \ - \ n}$$
• The Law of the Zero Exponent is as follows: $$a^0 \ = \ 1$$
• The Law of the Negative Exponent states that $$a^{-m} \ = \ \frac{1}{a^m}$$
• The Law of the Power of a Power is as follows: $$(a^m)^n \ = \ a^{m \times n}$$
• The Power of a Product is defined as $$(ab)^m \ = \ a^m \times b^m$$
• The Power of a Quotient is defined as $$(\frac{a}{b})^m \ = \ \frac{a^m}{b^m}$$

### Division Properties of Exponents

Some division properties of exponents are:

• The Law of Quotients states that $$\frac{a^m}{a^n} \ = \ a^{m \ - \ n}$$
• The Power of a Quotient is defined as $$(\frac{a}{b})^m \ = \ \frac{a^m}{b^m}$$

### Exercises for Division Property of Exponents

1) $$\frac{22x^7}{2x^3} =$$

2) $$\frac{9^5}{9} =$$

3) $$\frac{6x^6}{2x^3} =$$

4) $$\frac{24x^{-7}}{3x^{-3}} =$$

5) $$\frac{18x^{-6}}{2x^{-2}} =$$

6) $$\frac{16x^6}{4x^3} =$$

7) $$\frac{2^6}{2} =$$

8) $$\frac{10x^{-8}}{2x^{-3}y^{2}} =$$

9) $$\frac{18x^{-7}}{3x^{-4}y^{3}} =$$

10) $$\frac{3^5}{3} =$$

1) $$\frac{22x^7}{2x^3} =$$$$\ \color{red}{\frac{22}{2} \ x^{(7 - 3)}}$$$$\ \color{red}{= 11x^{4}}$$
2) $$\frac{9^5}{9} =$$$$\ \color{red}{9^{(5 - 1)}}$$$$\ \color{red}{ = 9^{4}}$$
3) $$\frac{6x^6}{2x^3} =$$$$\ \color{red}{\frac{6}{2} \ x^{(6 - 3)}}$$$$\ \color{red}{= 3x^{3}}$$
4) $$\frac{24x^{-7}}{3x^{-3}} =$$$$\ \color{red}{\frac{24}{3} \ x^{(-7 - (-3))}}$$$$\ \color{red}{= \frac{8}{x^{4}}}$$
5) $$\frac{18x^{-6}}{2x^{-2}} =$$$$\ \color{red}{\frac{18}{2} \ x^{(-6 - (-2))}}$$$$\ \color{red}{= \frac{9}{x^{4}}}$$
6) $$\frac{16x^6}{4x^3} =$$$$\ \color{red}{\frac{16}{4} \ x^{(6 - 3)}}$$$$\ \color{red}{= 4x^{3}}$$
7) $$\frac{2^6}{2} =$$$$\ \color{red}{2^{(6 - 1)}}$$$$\ \color{red}{ = 2^{5}}$$
8) $$\frac{10x^{-8}}{2x^{-3}y^{2}} =$$$$\ \color{red}{\frac{10}{2} \ x^{(-8 - (-3))}y^{-2}}$$$$\ \color{red}{= \frac{5}{x^{5}y^{2}}}$$
9) $$\frac{18x^{-7}}{3x^{-4}y^{3}} =$$$$\ \color{red}{\frac{18}{3} \ x^{(-7 - (-4))}y^{-3}}$$$$\ \color{red}{= \frac{6}{x^{3}y^{3}}}$$
10) $$\frac{3^5}{3} =$$$$\ \color{red}{3^{(5 - 1)}}$$$$\ \color{red}{ = 3^{4}}$$

## Division Property of Exponents Quiz

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