Overview
This lesson covers how to divide and simplify expressions involving radicals, including rationalizing denominators and handling variables with exponents.
Dividing and Simplifying Square Roots
- The square root of 64 is 8; the square root of x² is absolute value of x (|x|).
- When dividing square roots, first simplify the coefficients and cancel matching variables.
- Example: √(50x)/√(2x) = √(25) = 5.
Dividing and Simplifying Cube Roots
- The cube root of 27 is 3; the cube root of 8 is 2.
- For exponents: Divide the exponent by the index (e.g., 9/3 = 3 for a⁹).
- Example: ∛(27a⁹) / ∛(8b⁶) = 3a³ / 2b².
Complex Radicals with Variables
- Split numbers into their radical components to simplify.
- The square root of x⁸ is x⁴; the square root of y¹⁰ is |y⁵| (use absolute value for odd results).
- Cancel shared factors between numerator and denominator.
- Rationalize denominators by multiplying numerator and denominator by needed radical terms.
Handling Exponents
- When dividing like bases, subtract exponents.
- For odd exponents under square roots, use absolute value notation.
Rationalizing Denominators
- Multiply the numerator and denominator by a radical that eliminates the radical at the bottom.
- For cube roots or higher, use appropriate powers to clear the radical.
Advanced Example with Negative Exponents
- Subtract exponents for like bases; negative exponent in denominator becomes positive in numerator and vice versa.
- Rationalize higher index roots by multiplying by the radical with enough power to clear the denominator.
Example with Fourth Roots
- For roots higher than square, divide exponents by the index.
- Rationalize denominators by multiplying by the necessary power of the radical.
Key Terms & Definitions
- Radical — an expression using a root, such as square root (√) or cube root (∛).
- Rationalizing the denominator — eliminating radicals from the denominator of a fraction.
- Absolute value — written |x|, represents the non-negative value of x.
- Index — the degree of the root (2 for square root, 3 for cube root, etc.).
Action Items / Next Steps
- Practice dividing and simplifying radicals with variables and coefficients.
- Work on rationalizing denominators for different root indices.