Question

Radicals & Rational Exponents Quick Check

Answer 1

Sure! Here’s a quick check on radicals and rational exponents:

### Quick Check on Radicals and Rational Exponents

**1. Simplifying Radicals:**
- Simplify \( \sqrt{50} \).
- Solution:
\[
\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}
\]

**2. Simplifying Rational Exponents:**
- Simplify \( 16^{\frac{3}{4}} \).
- Solution:
\[
16^{\frac{3}{4}} = (16^{\frac{1}{4}})^3 = 2^3 = 8
\]

**3. Converting between Radicals and Rational Exponents:**
- Write \( \sqrt[3]{x^5} \) as a rational exponent.
- Solution:
\[
\sqrt[3]{x^5} = x^{\frac{5}{3}}
\]

**4. Operations with Radicals:**
- Add \( \sqrt{18} + \sqrt{8} \).
- Solution:
\[
\sqrt{18} = 3\sqrt{2}, \quad \sqrt{8} = 2\sqrt{2} \implies \sqrt{18} + \sqrt{8} = 3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}
\]

**5. Solving Equations with Rational Exponents:**
- Solve \( x^{\frac{2}{3}} = 9 \).
- Solution:
\[
x = 9^{\frac{3}{2}} = \sqrt{9^3} = \sqrt{729} = 27
\]

**6. Evaluating Expressions with Radicals:**
- Evaluate \( \sqrt{64} \times \sqrt{16} \).
- Solution:
\[
\sqrt{64} = 8, \quad \sqrt{16} = 4 \implies \sqrt{64} \times \sqrt{16} = 8 \times 4 = 32
\]

### Practice Problems
Try solving these on your own!

1. Simplify \( \sqrt{72} \).
2. Solve \( x^{\frac{1}{2}} = 25 \).
3. Convert \( y^{\frac{3}{5}} \) to radical form.
4. Simplify \( 27^{\frac{2}{3}} \).

Let me know if you'd like to discuss the solutions or concepts further!