Compute \( b^x \) for any base \( b>0,\ b\neq 1 \). Understand exponent rules through interactive activities.
\[ \boxed{b^x \cdot b^y = b^{\,x+y}} \qquad \boxed{\dfrac{b^x}{b^y} = b^{\,x-y}} \qquad \boxed{b^{-x} = \dfrac{1}{b^x}} \]
where \( b > 0,\ b \neq 1 \) and \( x,y \in \mathbb{R} \). Use the calculator below to verify these properties instantly.
Enter base and exponent, then click Calculate
Base must be positive and not equal to 1. Use "e" for Euler's number.
Choose a base \( b \) above. Compare \( b^x \cdot b^y \) with \( b^{x+y} \), and \( \dfrac{b^x}{b^y} \) with \( b^{x-y} \). The table updates when you click Calculate.
| \( x \) | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|
| \( y \) | 3 | 4 | 5 | 6 | 7 |
| \( b^x \) | — | — | — | — | — |
| \( b^y \) | — | — | — | — | — |
| \( \color{red}{b^x \cdot b^y} \) | — | — | — | — | — |
| \( \color{red}{b^{x+y}} \) | — | — | — | — | — |
| \( \color{blue}{\dfrac{b^x}{b^y}} \) | — | — | — | — | — |
| \( \color{blue}{b^{x-y}} \) | — | — | — | — | — |
Tip: The red columns should match, and the blue columns should match — demonstrating the product and quotient rules.
Verify \( b^{-x} = \dfrac{1}{b^x} \).
| \( x \) | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|
| \( b \) | e | 2 | 3 | 4 | 5 |
| \( b^{x} \) | — | — | — | — | — |
| \( \dfrac{1}{b^x} \) | — | — | — | — | — |
| \( b^{-x} \) | — | — | — | — | — |
For fixed \( x = 3 \), compare \( b^3 \cdot b^y \) with \( b^{3+y} \).
| \( y \) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \( b^3 \cdot b^y \) | — | — | — | — | — |
| \( b^{3+y} \) | — | — | — | — | — |