Quizzes Math
What is an exponent?
An exponent is a way to show how many times a number (called the base) is multiplied by itself.
For example:
$2^3 = 2 \times 2 \times 2 = 8$
Exponential Expression
An exponential expression has a base and an exponent.
It can be written as:
$ a^n $
Where:
- $a$ is the base
- $n$ is the exponent
This means you multiply $a$ by itself $n$ times.
Displayed version (inline format):
$a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}$
Example with Division of Exponents
If we divide powers with the same base:
$\frac{5^4}{5^2} = 5^{4-2} = 5^2 = 25$
Exponents with 0
Any non-zero number raised to the power of 0 is equal to 1.
Rule:
$a^0 = 1$, for all $a \ne 0$
Example:
$3^0 = 1$
$100^0 = 1$
$(-5)^0 = 1$
⚠️ Exception: $0^0$ is undefined.
Exponents with 1
Any number raised to the power of 1 is equal to itself.
Rule:
$a^1 = a$
Example:
$7^1 = 7$
$(-2)^1 = -2$
Adding Exponents (Product Rule)
Rules :$a^m \times a^n=a^{m+n}$
If the base is the same, multiply by adding the exponents:
Formula: $a^m \times a^n=a^{m+n}$
Exemples :
- $2^3 \times 2^4=2^{3+4}=2^7=128$
- $5^2 \times 5^3=5^{2+3}=5^5=3125$
- $10^1 \times 10^2=10^3=1000$
Subtracting Exponents (Quotient Rule)
Rule:
If the base is the same, subtract the exponents when dividing:
$$
\frac{a^m}{a^n}=a^{m-n}, \quad \text { as long as } a \neq 0
$$
Examples:
- $\frac{2^5}{2^3}=2^2=4$
- $\frac{10^4}{10^1}=10^3=1000$
- ${\frac{7}{} 7^3}^3=7^0=1$
Multiplying Different Bases (Same Exponent)
Rule:
If the exponent is the same, multiply the bases and keep the exponent:
$$
a^n \times b^n=(a \times b)^n
$$
Examples:
- $2^3 \times 3^3=(2 \times 3)^3=6^3=216$
- $5^2 \times 4^2=(5 \times 4)^2=20^2=400$
Dividing Different Bases (Same Exponent)
Rule:
If the exponent is the same, divide the bases and keep the exponent:
$$
\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n, \quad \text { as long as } b \neq 0
$$
Examples:
- $\frac{6^2}{3^2}=\left(\frac{6}{3}\right)^2=2^2=4$
- $\frac{10^3}{2^3}=\left(\frac{10}{2}\right)^3=5^3=125$