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 |
| \( b^{x} \) | |||||
| \( \dfrac{1}{b^x} \) | |||||
| \( b^{-x} \) |