Powers with Integer Exponents
Doubling with a Look into the Past
Powers with Integer Exponents
Example: Doubling with a Look into the Past
Starting from our example:
| Time (in min.) \(= t\) | Number of Cents \(= C\) | Powers (new notation) |
|---|---|---|
| 0 | 1 | \(2^{0}\) |
| 1 | 2 | \(2^{1}\) |
| 2 | \(2 \cdot 2 = 4\) | \(2^{2}\) |
| 3 | \(2 \cdot 2 \cdot 2 = 8\) | \(2^{3}\) |
| \(\ldots\) | \(\ldots\) | \(\ldots\) |
Let’s take a look into the past. If I have one cent at starting time “0”, how many cents did I have one minute ago, two minutes ago, …?
| Time (in min.) \(= t\) | Number of Cents \(= C\) | Powers (new notation) |
|---|---|---|
| \(-4\) (4 minutes ago) | \(\frac{1}{2^{4}} = \frac{1}{16}\) | \(2^{-4}\) |
| \(-3\) (3 minutes ago) | \(\frac{1}{2^{3}} = \frac{1}{8}\) | \(2^{-3}\) |
| \(-2\) (2 minutes ago) | \(\frac{1}{2^{2}} = \frac{1}{4}\) | \(2^{-2}\) |
| \(-1\) (1 minute ago) | \(\frac{1}{2^{1}} = \frac{1}{2}\) | \(2^{-1}\) |
| 0 | 1 | \(2^{0}\) |
| 1 | 2 | \(2^{1}\) |
| 2 | \(2 \cdot 2 = 4\) | \(2^{2}\) |
| 3 | \(2 \cdot 2 \cdot 2 = 8\) | \(2^{3}\) |
| \(\ldots\) | \(\ldots\) | \(\ldots\) |
To describe growth uniformly, one defines:
\(2^{-1} = \frac{1}{2^{1}}; \quad 2^{-2} = \frac{1}{2^{2}}; \quad 2^{-3} = \frac{1}{2^{3}}, \ldots\)
Definition: Powers with Negative Integer Exponents
For all rational (also real) numbers \(a \neq 0\) and natural numbers \(n \in \mathbb{N}\): \[a^{-n} = \frac{1}{a^{n}}\]
Note: \(a \neq 0\), because division by zero is not allowed!
Exercises (without calculator)
Write as a power of ten (with negative exponents)
- \(0.1 = \frac{1}{10} = \frac{1}{10^{1}} = 10^{-1}\)
- \(0.01 =\)
- \(0.001 =\)
Write as a decimal number without powers
- \(7 \cdot 10^{-3} = 7 \cdot 0.001 = 0.007\)
- \(1.2 \cdot 10^{-4} =\)
- \(10^{-7} =\)
Write as a fraction without powers
- \(5^{-2} = \frac{1}{5^{2}} = \frac{1}{25}\)
- \(\left(\frac{2}{5}\right)^{-3} =\)
- \((\sqrt{2})^{-2} =\)
- \((-4)^{-3} =\)