Overview
This lesson covers how to simplify radicals, including real and imaginary square roots, combining like radicals, and rationalizing denominators in radical expressions.
Simplifying Basic Radicals
- The square root of a perfect square (e.g., √36) is a whole number (√36 = 6).
- The square root of a negative number (e.g., √-49) results in an imaginary number (√-49 = 7i, where i = √-1).
- A negative outside the radical affects only the sign (e.g., -√64 = -8).
- For -√-25, separate terms: √25 = 5, √-1 = i, final answer is -5i.
Recognizing Perfect Squares
- Perfect squares up to 10: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
- Knowing more perfect squares (up to 20² = 400) is helpful.
Simplifying Non-Perfect Squares
- Break down the radicand into a product of a perfect square and another factor: √75 = √25 × √3 = 5√3.
- Choose the largest perfect square factor for simplification.
Practice Examples
- √18 = √9 × √2 = 3√2.
- √48 = √16 × √3 = 4√3.
- 8√80 = 8 × √16 × √5 = 8 × 4√5 = 32√5.
- 5√98 = 5 × √49 × √2 = 5 × 7√2 = 35√2.
Rationalizing Denominators
- To rationalize 5/√2, multiply numerator and denominator by √2: (5√2)/2.
- For 1/√3, multiply by √3: (√3)/3.
- For 1/√5, multiply by √5: (√5)/5.
Adding and Subtracting Radicals
- Simplify each radical before combining like terms.
- Example: 4√8 + 3√50 - 6√32 = 8√2 + 15√2 - 24√2 = -√2.
- Like radicals can be combined by adding/subtracting their coefficients.
Combining Radicals with Different Radicands
- 7√27 + 3√12 - 5√48 simplifies to 21√3 + 6√3 - 20√3 = 7√3.
Rationalizing with Conjugates
- For 8/(3-√2), multiply numerator and denominator by (3+√2): result is (24 + 8√2)/7.
- For (3+√2)/(5-√2), multiply by (5+√2): result is (17 + 8√2)/23.
Key Terms & Definitions
- Radical — An expression that includes a root, such as √x.
- Imaginary Number — A number that involves i, where i = √-1.
- Perfect Square — A number that is the square of an integer.
- Rationalize — To eliminate radicals from the denominator of a fraction.
- Conjugate — For (a - b), the conjugate is (a + b).
Action Items / Next Steps
- Practice simplifying and combining radicals with and without coefficients.
- Memorize perfect squares up to at least 20².
- Complete assigned problems on rationalizing denominators and adding/subtracting radicals.