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8η τάξη
Course: 8η τάξη > Ενότητα 1
Μάθημα 6: Αρνητικοί εκθέτεςNegative exponents review
Review the basics of negative exponents and try some practice problems.
Definition for negative exponents
We define a negative power as the multiplicative inverse of the base raised to the positive opposite of the power:
Want to learn more about this definition? Check out this video.
Παραδείγματα:
- 3, start superscript, minus, 5, end superscript, equals, start fraction, 1, divided by, 3, start superscript, 5, end superscript, end fraction
- start fraction, 1, divided by, 2, start superscript, 8, end superscript, end fraction, equals, 2, start superscript, minus, 8, end superscript
- y, start superscript, minus, 2, end superscript, equals, start fraction, 1, divided by, y, squared, end fraction
- left parenthesis, start fraction, 8, divided by, 6, end fraction, right parenthesis, start superscript, minus, 3, end superscript, equals, left parenthesis, start fraction, 6, divided by, 8, end fraction, right parenthesis, cubed
Some intuition
So why do we define negative exponents this way? Here are a couple of justifications:
Justification #1: Patterns
n | 2, start superscript, n, end superscript |
---|---|
3 | 2, cubed, equals, 8 |
2 | 2, squared, equals, 4 |
1 | 2, start superscript, 1, end superscript, equals, 2 |
0 | 2, start superscript, 0, end superscript, equals, 1 |
minus, 1 | 2, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 2, end fraction |
minus, 2 | 2, start superscript, minus, 2, end superscript, equals, start fraction, 1, divided by, 4, end fraction |
Notice how 2, start superscript, n, end superscript is divided by 2 each time we reduce n. This pattern continues even when n is zero or negative.
Justification #2: Exponent properties
Recall that start fraction, x, start superscript, n, end superscript, divided by, x, start superscript, m, end superscript, end fraction, equals, x, start superscript, n, minus, m, end superscript. So...
We also know that
And so we get 2, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 2, end fraction.
Also, recall that x, start superscript, n, end superscript, dot, x, start superscript, m, end superscript, equals, x, start superscript, n, plus, m, end superscript. So...
And indeed, according to the definition...
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