Exponent Properties

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 are integers

3. $(a^m)^n = a^{m \times n}$, where $a \neq 0, m, n$ are integers

4. $(ab)^m = a^m \times b^m$, where $a, b \neq 0$ , and is an integer

5. $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$, where , and is an integer

6. $(a^\frac{m}{n})^p \cdot (a^\frac{p}{n}) = (a)^\frac{m+p}{n}$
7. $(a^\frac{m}{n}) \cdot (a^\frac{p}{q}) = (a)^\frac{m \cdot q + p \cdot n}{n \cdot q}$

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.

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 $\frac{2^5 \times 2^3}{2^2}$

   $$\frac{2^5 \times 2^3}{2^2} = \frac{2^{5+3}}{2^2} = \frac{2^8}{2^2} = 2^{8-2} = 2^6 = 64$$

2. Simplify $2^2 \cdot 2^3$

   $$2^2 \cdot 2^3 = 2^{2+3} = 2^5 = 32$$

3. Simplify $2^5 \cdot 2^2$

   $$2^5 \cdot 2^2 = 2^{5+2} = 2^7 = 128$$

4. Simplify $2^3 \cdot 2^7$

   $$2^3 \cdot 2^7 = 2^{3+7} = 2^{10} = 1024$$

5. Simplify $\frac{2^8}{2^6}$

   $$\frac{2^8}{2^6} = 2^{8-6} = 2^2 = 4$$

6. Simplify $\frac{2^{10}}{2^3}$

   $$\frac{2^{10}}{2^3} = 2^{10-3} = 2^7 = 128$$

7. Simplify $\frac{2^6}{2^4}$

   $$\frac{2^6}{2^4} = 2^{6-4} = 2^2 = 4$$

8. Simplify $(2^3)^3$

   $$(2^3)^3 = 2^{3 \times 3} = 2^9 = 512$$

9. Simplify $(2^4)^2$

   $$(2^4)^2 = 2^{4 \times 2} = 2^8 = 256$$

10. Simplify $(2^2)^5$

    $$(2^2)^5 = 2^{2 \times 5} = 2^{10} = 1024$$

Dividing two exponents with the same base results in a new exponent with the difference of the powers.

An exponent of an exponent means multiplying the power by the outer power.

The exponent of a multiplication equals the multiplication of each base raised to the same power.

The exponent of a division equals the division of each base raised to the same power.

, where $a > 0$, $\frac{m}{n}$ and $\frac{p}{n}$ are rational numbers with $n \neq 0$

, where $a > 0$, $\frac{m}{n}$ and $\frac{p}{q}$ are rational numbers with $n, q \neq 0$