An easy to use calculator to compute exponential functions of the form \( b^x \) to any base \( b \) is presented. Activities related to the product and quotient of like bases exponential rules described below, using the calculator, are also included.
\( \)\( \)\( \)\( \)\( \)\( \)
The exponential function \( b^x \) is defined for \( b \gt 0 \) and \( b \ne 1 \)
1 - Product of Like Bases Rule
\[ b^x b^y = b^{x+y} \]
2 - Quotient of Like Bases Rule
\[ \dfrac{b^x}{b^y} = b^{x-y} \]
Answer
Activity 1: Product and Quotient of Like Bases Rules
Chose any base \( b \) and use the calculator to find the values of \( b^x \), \( b^y \), \( b^{x+y} \), \( b^x \cdot b^y \), \( \dfrac{b^x}{b^y} \) and \( b^{x-y} \)
a) Compare the quantites \( b^x \cdot b^y \) and \( b^{x+y} \) for each pair of values \( (x,y) \). These quantities are equal according to the product rule of like bases in 1) above.
b) Compare the quantites \( \dfrac{b^x}{b^y} \) and \( b^{x-y} \) for each pair of values \( (x,y) \). These quantities are equal according to the quotient of like bases rule in 2) above.
| \( x \) | 4 | 5 | 25 | 40 | 100 | 120 | 1000 |
| \( b^x \) | |||||||
| \( y \) | 2 | 4 | 5 | 10 | 25 | 60 | 100 |
| \( b^y \) | |||||||
| \( \color{red}{b^x \cdot b^y} \) | |||||||
| \( \color{red}{b^{x+y}} \) | |||||||
| \( \color{blue}{\dfrac{b^x}{b^y}} \) | |||||||
| \( \color{blue}{b^{x-y}} \) |
Activity 2: Negative Exponents
The negative exponent is defined as follows
\[ b^{-x} = \dfrac{1}{b^x} \]
Use the calculator to calculate \( b^x \) and \( b^{-x} \) and compare the quantities \( b^{-x} \) and \( \dfrac{1}{b^x} \) which according to the definition above are equal.
| \( x \) | 4 | 5 | 25 | 40 | 100 |
| \( b \) | e | 3 | 4 | 5 | 10 |
| \( b^{x} \) | |||||
| \( \dfrac{1}{b^x} \) | |||||
| \( b^{-x} \) |
