Question

Complete the table. For each expression, choose which exponent rule has been used to simplify. 11^80/11^22 = 11^58. 6^-12/9^-15 - 9^15/6^12, (16^3)^9 = 16^27. Assign each of these to either product rule, quotient rule, power rule, negative integer exponent rule, or zero power rule.

Answer 1

Sure! Here’s the breakdown of the exponent rules used in each expression:

1. **Expression**: \( \frac{11^{80}}{11^{22}} = 11^{58} \)
- **Exponent Rule**: **Quotient Rule**
- **Explanation**: The quotient rule states that when you divide two powers with the same base, you subtract the exponents: \( a^m / a^n = a^{m-n} \).

2. **Expression**: \( 6^{-12}/9^{-15} - 9^{15}/6^{12} \)
- **Exponent Rule**: **Negative Integer Exponent Rule** (applied to convert negative exponents into positive)
- **Explanation**: The negative integer exponent rule states that \( a^{-n} = \frac{1}{a^n} \). This is used to rewrite the terms with negative exponents.

3. **Expression**: \( (16^3)^9 = 16^{27} \)
- **Exponent Rule**: **Power Rule**
- **Explanation**: The power rule states that when raising a power to a power, you multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).

So the completed table would look like this:

| Expression | Exponent Rule |
|---------------------------------------------------|-----------------------------------|
| \( \frac{11^{80}}{11^{22}} = 11^{58} \) | Quotient Rule |
| \( \frac{6^{-12}}{9^{-15}} - \frac{9^{15}}{6^{12}} \) | Negative Integer Exponent Rule |
| \( (16^3)^9 = 16^{27} \) | Power Rule |