Negative exponents represent the reciprocal of a number raised to the positive version of that exponent. In other words, a number raised to a negative exponent is equivalent to 1 divided by that number raised to the positive exponent. ...
For example: a^-2 = 1/a^2
Simplifying the Expression
Let's break down the simplification of the expression (8a^6)(-2a^-9):
- Separate the numerical coefficients and the variable terms: (8)(-2) (a^6)(a^-9)
- Multiply the coefficients: (8)(-2) = -16
- Apply the rule of exponents when multiplying terms with the same base: a^m * a^n = a^(m+n)
- Combine the variable terms: a^6 * a^-9 = a^(6+(-9)) = a^-3
- Rewrite the negative exponent as a positive exponent: a^-3 = 1/a^3
- Combine the results: -16 * a^-3 = -16 * (1/a^3) = -16/a^3
Final Result
Therefore, the simplified expression with only positive exponents is **-16/a^3**.
Key Takeaways
This example highlights the following key points:
- Negative exponents represent reciprocals.
- When multiplying terms with the same base, add the exponents.
- Expressions can be simplified by applying these rules.
Additional Considerations
It's important to note that the final expression -16/a^3 is undefined if a = 0. This is because division by zero is undefined in mathematics.
Practice Problems
Here are some practice problems to solidify your understanding of simplifying expressions with negative exponents:
- (3x^4) (-5x^-2)
- (2y^-3) (4y^5)
- (7z^2) (z^-6)
Remember to apply the rules we discussed and express the final results with only positive exponents.