Question

rational exponents quick check

Answer 1

Rational exponents are a way to express roots in exponential form. The general rule for rational exponents is:

\[
a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m
\]

Here's a quick summary of how to work with rational exponents:

1. **Converting to Radical Form**:
- If you have \( a^{\frac{1}{n}} \), this is the \( n \)-th root of \( a \), or \( \sqrt[n]{a} \).
- If you have \( a^{\frac{m}{n}} \), this means \( \sqrt[n]{a^m} \) or \( (\sqrt[n]{a})^m \).

2. **Examples**:
- \( 8^{\frac{1}{3}} = \sqrt[3]{8} = 2 \) (since \( 2^3 = 8 \))
- \( 27^{\frac{2}{3}} = \sqrt[3]{27^2} = \sqrt[3]{729} = 9 \) (since \( 9^3 = 729 \))

3. **Multiplying Bases with Rational Exponents**:
- \( a^{m/n} \cdot a^{p/q} = a^{\frac{mq + np}{nq}} \)

4. **Dividing Bases with Rational Exponents**:
- \( \frac{a^{m/n}}{a^{p/q}} = a^{\frac{mq - np}{nq}} \)

5. **Power of a Power**:
- \( (a^{m/n})^p = a^{\frac{mp}{n}} \)

6. **Negative Exponents**:
- \( a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} \)

### Quick Check Questions:
1. Convert \( x^{\frac{3}{2}} \) to radical form.
2. Simplify \( (16^{\frac{1}{2}})^3 \).
3. Evaluate \( 64^{\frac{2}{3}} \).
4. Multiply: \( a^{\frac{1}{2}} \cdot a^{\frac{1}{3}} \).
5. Divide: \( b^{\frac{5}{6}} / b^{\frac{1}{2}} \).

### Answers:
1. \( x^{\frac{3}{2}} = \sqrt{x^3} = \sqrt{x^3} \)
2. \( (16^{\frac{1}{2}})^3 = 4^3 = 64 \)
3. \( 64^{\frac{2}{3}} = \sqrt[3]{64^2} = \sqrt[3]{4096} = 16 \)
4. \( a^{\frac{1}{2}} \cdot a^{\frac{1}{3}} = a^{\frac{3 + 2}{6}} = a^{\frac{5}{6}} \)
5. \( b^{\frac{5}{6}} / b^{\frac{1}{2}} = b^{\frac{5 - 3}{6}} = b^{\frac{2}{6}} = b^{\frac{1}{3}} \)

Feel free to ask if you have any specific questions or need further clarification!