# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/exponential-logarithm/properties Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/exponential-logarithm/properties/en.mdx Learn 7 fundamental exponent rules with practical examples. Learn multiplication, division, power operations and rational exponents for problem solving. --- ## Table of Exponent Values for Base Two | $$2^n$$ | Exponentiation Result | | ---------------------------- | --------------------- | | $$2^1$$ | $$2$$ | | $$2^2$$ | $$4$$ | | $$2^3$$ | $$8$$ | | $$2^4$$ | $$16$$ | | $$2^5$$ | $$32$$ | | $$2^6$$ | $$64$$ | | $$2^7$$ | $$128$$ | | $$2^8$$ | $$256$$ | | $$2^9$$ | $$512$$ | | $$2^{10}$$ | $$1024$$ | Visible text: | | Exponentiation Result | | ---------------------------- | --------------------- | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ## Exponent Properties There are several exponent properties that need to be understood: 1. $$a^m \cdot a^n = a^{m+n}$$, where $$a \neq 0, m, n$$ are integers This means multiplying two exponents with the same base results in a new exponent with the sum of the powers. 2. $$\frac{a^m}{a^n} = a^{m-n}$$, where $$a \neq 0, m, n$$ are integers Dividing two exponents with the same base results in a new exponent with the difference of the powers. 3. $$(a^m)^n = a^{m \times n}$$, where $$a \neq 0, m, n$$ are integers An exponent of an exponent means multiplying the power by the outer power. 4. $$(ab)^m = a^m \times b^m$$, where $$a, b \neq 0$$ , and $$m$$ is an integer The exponent of a multiplication equals the multiplication of each base raised to the same power. 5. $$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$$, where $$b \neq 0$$ , and $$m$$ is an integer The exponent of a division equals the division of each base raised to the same power. 6. $$(a^\frac{m}{n})^p \cdot (a^\frac{p}{n}) = (a)^\frac{m+p}{n}$$ , where $$a > 0$$, $$\frac{m}{n}$$ and $$\frac{p}{n}$$ are rational numbers with $$n \neq 0$$ 7. $$(a^\frac{m}{n}) \cdot (a^\frac{p}{q}) = (a)^\frac{m \cdot q + p \cdot n}{n \cdot q}$$ , where $$a > 0$$, $$\frac{m}{n}$$ and $$\frac{p}{q}$$ are rational numbers with $$n, q \neq 0$$ Visible text: 1. , where are integers This means multiplying two exponents with the same base results in a new exponent with the sum of the powers. 2. , where are integers Dividing two exponents with the same base results in a new exponent with the difference of the powers. 3. , where are integers An exponent of an exponent means multiplying the power by the outer power. 4. , where , and is an integer The exponent of a multiplication equals the multiplication of each base raised to the same power. 5. , where , and is an integer The exponent of a division equals the division of each base raised to the same power. 6. , where , and are rational numbers with 7. , where , and are rational numbers with ## Importance of Conditions for Each Property Each exponent property has specific conditions: - In properties $$1$$, $$2$$, and $$3$$, the value $$a \neq 0$$ because exponents with base $$0$$ are only defined for positive powers. - In property $$4$$, the values $$a, b \neq 0$$ to ensure the exponent is defined. - In property $$5$$, the value $$b \neq 0$$ to avoid division by zero. - In properties $$6$$ and $$7$$, the value $$a > 0$$ because rational exponents on negative numbers can result in complex numbers. Visible text: - In properties , , and , the value because exponents with base are only defined for positive powers. - In property , the values to ensure the exponent is defined. - In property , the value to avoid division by zero. - In properties and , the value because rational exponents on negative numbers can result in complex numbers. Understanding these exponent properties is very important as a foundation for advanced mathematics learning, such as logarithms, exponential functions, and calculus. ## Worked Examples 1. **Simplify $$\frac{2^5 \times 2^3}{2^2}$$** ```math \frac{2^5 \times 2^3}{2^2} = \frac{2^{5+3}}{2^2} = \frac{2^8}{2^2} = 2^{8-2} = 2^6 = 64 ``` 2. **Simplify $$2^2 \cdot 2^3$$** ```math 2^2 \cdot 2^3 = 2^{2+3} = 2^5 = 32 ``` 3. **Simplify $$2^5 \cdot 2^2$$** ```math 2^5 \cdot 2^2 = 2^{5+2} = 2^7 = 128 ``` 4. **Simplify $$2^3 \cdot 2^7$$** ```math 2^3 \cdot 2^7 = 2^{3+7} = 2^{10} = 1024 ``` 5. **Simplify $$\frac{2^8}{2^6}$$** ```math \frac{2^8}{2^6} = 2^{8-6} = 2^2 = 4 ``` 6. **Simplify $$\frac{2^{10}}{2^3}$$** ```math \frac{2^{10}}{2^3} = 2^{10-3} = 2^7 = 128 ``` 7. **Simplify $$\frac{2^6}{2^4}$$** ```math \frac{2^6}{2^4} = 2^{6-4} = 2^2 = 4 ``` 8. **Simplify $$(2^3)^3$$** ```math (2^3)^3 = 2^{3 \times 3} = 2^9 = 512 ``` 9. **Simplify $$(2^4)^2$$** ```math (2^4)^2 = 2^{4 \times 2} = 2^8 = 256 ``` 10. **Simplify $$(2^2)^5$$** ```math (2^2)^5 = 2^{2 \times 5} = 2^{10} = 1024 ``` Visible text: 1. **Simplify ** 2. **Simplify ** 3. **Simplify ** 4. **Simplify ** 5. **Simplify ** 6. **Simplify ** 7. **Simplify ** 8. **Simplify ** 9. **Simplify ** 10. **Simplify **