This page is designed to help students, parents, and teachers master the properties of exponents through carefully selected questions and detailed step-by-step solutions available immediately under each question block.
The exponential form is a convenient way to write long, repeated multiplications of the same number by itself:
\[ \underbrace{ a \cdot a \cdot a \dots a}_{ n \text{ times} } = a^n \]
Here, \( a \) is called the base (a real number) and \( n \) is called the exponent (an integer). The expression \( a^n \) is read as "\(a\) to the power of \(n\)." Understanding these rules is a critical stepping stone for advanced algebra and calculus.
Review these fundamental properties before attempting the questions below:
| # | Rule Name | Definition | Examples |
|---|---|---|---|
| 1 | Exponent Form | \( \underbrace{ a \cdot a \cdot a \dots a}_{ n \text{ times} } = a^n \) | \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 4^5 \) \( 2^3 = 2 \cdot 2 \cdot 2 = 8 \) |
| 2 | Negative Exponent | \( a^{-n} = \dfrac{1}{a^n} \) or \( a^{n} = \dfrac{1}{a^{-n}} \) | \( 3^{-4} = \dfrac{1}{3^4} \) \( 5^{-2} = \dfrac{1}{5^2} \) |
| 3 | Product Rule (Same Base) | \( a^m \cdot a^n = a^{m+n} \) | \( 2^4 \cdot 2^6 = 2^{4+6} = 2^{10} \) \( 3^{2+6} = 3^2 \cdot 3^6 \) |
| 4 | Product Rule (Same Exponent) | \( a^m \cdot b^m = (a \cdot b)^m \) | \( 2^5 \cdot 3^5 = (2 \cdot 3)^5 = 6^5 \) \( (4 \cdot 3)^2 = 4^2 \cdot 3^2 \) |
| 5 | Quotient Rule (Same Base) | \( \dfrac{a^m}{a^n} = a^{m - n} \) | \( \dfrac{2^6}{2^4} = 2^{6-4} = 2^{2} \) \( 3^{5-2} = \dfrac{3^5}{3^2} \) |
| 6 | Quotient Rule (Same Exponent) | \( \left( \dfrac{a}{b} \right)^m = \dfrac{a^m}{b^m} \) | \( \left( \dfrac{3}{5} \right)^4 = \dfrac{3^4}{5^4} \) \( \dfrac{4^2}{5^2} = \left( \dfrac{4}{5} \right)^2 \) |
| 7 | Quotient Rule (Negative Exponent) | \( \left( \dfrac{a}{b} \right)^{-m} = \dfrac{b^m}{a^m} \) | \( \left( \dfrac{3}{5} \right)^{-2} = \dfrac{5^2}{3^2} \) |
| 8 | Power Rule | \( (a^n)^m = a^{n \cdot m} \) | \( (2^3)^4 = 2^{3\cdot4} = 2^{12} \) \( 3^{4 \cdot 5} = (3^4)^5 = (3^5)^4 \) |
| 9 | Zero Exponent Rule | \( a^0 = 1 \) (for \(a \neq 0\)) | \( 10000000^0 = 1 \) Note: \( \color{red}{0^0 \text{ is undefined}} \) |
| 10 | Exponent One Rule | \( a^1 = a \) | \( 45^1 = 45 \) \( 100 = 100^1 \) |
| 11 | Base One Rule | \( 1^n = 1 \) | \( 1^{230} = 1 \) \( 1^{-100} = 1 \) |
| 12 | Negative One Base Rule | \( (-1)^n = \begin{cases} 1, & \text{if } n \text{ is even} \\ -1, & \text{if } n \text{ is odd} \end{cases} \) | \( (-1)^{19} = -1 \) \( (-1)^{18} = 1 \) |
Instructions: Try to solve all the following questions without using a calculator before expanding the solutions block.