## How to Multiply Exponents

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In mathematics, an exponent refers to the number of times a number, known as the base, is multiplied by itself. For example, in the expression  $$5^7$$,  $$5$$ is the base and is multiplied by itself  $$7$$ times, resulting in $$5 \ \times \ 5 \ \times \ 5 \ \times \ 5 \ \times \ 5 \ \times \ 5 \ \times \ 5$$. This format is a neat and time-saving way to represent repeated multiplication.
Now, let’s consider the same exponent $$5^7$$. Here we can say that this exponent has 2 parts. In this case, $$5$$ is referred to as the "base," while $$7$$ 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 $$(\frac{a}{b})^m \ = \ \frac{a^m}{b^m}$$

### Multiplication Properties of Exponents

Some properties of exponents multiplication are:

• 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 Power of a Power is as follows: $$(a^m)^n \ = \ a^{m \times n}$$
• The Power of a Product is defined as $$(ab)^m \ = \ a^m \times b^m$$

### Exercises for Multiplication Property of Exponents

1) $$10^3\ \times \ 10^7 =$$

2) $$9x^3 \ \times \ x^4 \ \times \ x =$$

3) $$2x^7 \ \times \ x^5 \ \times \ x =$$

4) $$6x^7 \ \times \ 4yx^4 =$$

5) $$7^4\ \times \ 7 =$$

6) $$(6x^2)^{2} =$$

7) $$8^3\ \times \ 8 =$$

8) $$7x^5 \ \times \ 4yx^7 =$$

9) $$8y^7x^2 \ \times \ 3y^4x^3 =$$

10) $$10x^8 \ \times \ x^4 \ \times \ x =$$

1) $$10^3\ \times \ 10^7 =$$$$\ \color{red}{10^{3 \ + \ 7}}$$$$\ \color{red}{= 10^{10}}$$
2) $$9x^3 \ \times \ x^4 \ \times \ x =$$$$\ \color{red}{9x^{3 \ + \ 4 \ + \ 1}}$$$$\ \color{red}{= 9^{8}}$$
3) $$2x^7 \ \times \ x^5 \ \times \ x =$$$$\ \color{red}{2x^{7 \ + \ 5 \ + \ 1}}$$$$\ \color{red}{= 2^{13}}$$
4) $$6x^7 \ \times \ 4yx^4 =$$$$\ \color{red}{(6 \ \times \ 4)yx^{(7 + 4)}}$$$$\ \color{red}{= 24yx^{11}}$$
5) $$7^4\ \times \ 7 =$$$$\ \color{red}{7^{4 \ + \ 1}}$$$$\ \color{red}{= 7^{5}}$$
6) $$(6x^2)^{2} =$$$$\ \color{red}{(6^2)x^{(2 \times 2)}}$$$$\ \color{red}{= 36x^{4}}$$
7) $$8^3\ \times \ 8 =$$$$\ \color{red}{8^{3 \ + \ 1}}$$$$\ \color{red}{= 8^{4}}$$
8) $$7x^5 \ \times \ 4yx^7 =$$$$\ \color{red}{(7 \ \times \ 4)yx^{(5 + 7)}}$$$$\ \color{red}{= 28yx^{12}}$$
9) $$8y^7x^2 \ \times \ 3y^4x^3 =$$$$\ \color{red}{(8 \times 3)y^{(7 + 4)}x^{(2 + 3)}}$$$$\ \color{red}{= 24y^{11}x^{5}}$$
10) $$10x^8 \ \times \ x^4 \ \times \ x =$$$$\ \color{red}{10x^{8 \ + \ 4 \ + \ 1}}$$$$\ \color{red}{= 10^{13}}$$

## Multiplication Property of Exponents Quiz

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