Solved Examples: Rationalizing Monomial Denominators

📚 Quick Study Guide

• 🔍 Rationalizing the Denominator: The process of eliminating radicals (like square roots) from the denominator of a fraction.
• 💡 Monomial Denominator: A denominator consisting of a single term (e.g., $3\sqrt{x}$, $5y^2$).
• 📝 Key Principle: Multiply both the numerator and denominator by a suitable expression that will eliminate the radical in the denominator.
• ➗ For Square Roots: If the denominator is $\sqrt{a}$, multiply both numerator and denominator by $\sqrt{a}$.
• ➕ For nth Roots: If the denominator is $\sqrt[n]{a^m}$, multiply both numerator and denominator by $\sqrt[n]{a^{n-m}}$.
• 🔢 Example: To rationalize $\frac{1}{\sqrt{2}}$, multiply by $\frac{\sqrt{2}}{\sqrt{2}}$ to get $\frac{\sqrt{2}}{2}$.
• 🧠 General Formula: $\frac{a}{\sqrt[n]{b^m}} = \frac{a}{\sqrt[n]{b^m}} \cdot \frac{\sqrt[n]{b^{n-m}}}{\sqrt[n]{b^{n-m}}} = \frac{a\sqrt[n]{b^{n-m}}}{b}$

Practice Quiz

1. Question 1: What is the first step in rationalizing a monomial denominator?
1. A. Simplify the numerator.
2. B. Identify the radical in the denominator.
3. C. Square both the numerator and denominator.
4. D. Add 1 to the denominator.
2. Question 2: Rationalize the denominator: $\frac{2}{\sqrt{3}}$
1. A. $\frac{2\sqrt{3}}{9}$
2. B. $\frac{\sqrt{3}}{3}$
3. C. $\frac{2\sqrt{3}}{3}$
4. D. $\sqrt{3}$
3. Question 3: Rationalize the denominator: $\frac{5}{\sqrt{5}}$
1. A. $\sqrt{5}$
2. B. $5\sqrt{5}$
3. C. $\frac{\sqrt{5}}{5}$
4. D. $25$
4. Question 4: Rationalize the denominator: $\frac{1}{\sqrt[3]{2}}$
1. A. $\frac{\sqrt[3]{4}}{2}$
2. B. $\frac{\sqrt[3]{2}}{2}$
3. C. $\frac{1}{2}$
4. D. $\sqrt[3]{4}$
5. Question 5: Rationalize the denominator: $\frac{4}{\sqrt[4]{8}}$
1. A. $2\sqrt[4]{2}$
2. B. $\sqrt[4]{2}$
3. C. $\frac{\sqrt[4]{2}}{2}$
4. D. $\frac{4\sqrt[4]{2}}{2}$
6. Question 6: Simplify $\frac{x}{\sqrt{x}}$ assuming x > 0.
1. A. $x$
2. B. $\sqrt{x}$
3. C. $x^2$
4. D. $1$
7. Question 7: Rationalize the denominator: $\frac{7}{\sqrt[5]{16}}$
1. A. $\frac{7\sqrt[5]{4}}{4}$
2. B. $\frac{\sqrt[5]{4}}{2}$
3. C. $\frac{7\sqrt[5]{16}}{4}$
4. D. $\frac{7\sqrt[5]{2}}{2}$