Simplifying Radicals with Variables and Exponents
Introduction
- Focus on simplifying radicals with variables and exponents.
- Example: Simplify ( \sqrt{x^5} ).
Simplifying Radicals
Method 1: Repeated Multiplication
- Write the variable multiple times (e.g., (x^5 = x \cdot x \cdot x \cdot x \cdot x)).
- Group terms based on the index (2 for square roots).
- ( \sqrt{x^5} = x^2 \cdot \sqrt{x} ).
Method 2: Division
- Determine how many times the index divides the exponent.
- Example: ( \sqrt{x^7} )
- 2 goes into 7 three times with a remainder of 1.
- Result: ( x^3 \cdot \sqrt{x} ).
Examples
- ( \sqrt{x^8} ):
- 2 goes into 8 four times, no remainder.
- Result: ( x^4 ).
- ( \sqrt{x^9} ):
- 2 goes into 9 four times with remainder 1.
- Result: ( x^4 \cdot \sqrt{x} ).
Simplifying Radicals with Numbers
Example: ( \sqrt{32} )
- Break into perfect square components: 32 = 16 ( \times ) 2.
- ( \sqrt{32} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} ).
Complex Examples
Example: ( \sqrt{50x^3y^{18}} )
- Simplify each component:
- ( \sqrt{50} = 5\sqrt{2} ).
- ( \sqrt{x^3} = x \sqrt{x} ).
- ( \sqrt{y^{18}} = y^9 ).
- Consider absolute values for even indexes resulting in odd exponents.
- Result: ( 5x y^9 \sqrt{2x} ) with absolute values for odd components.
Cube Roots and Higher Indexes
- Example: ( \sqrt[3]{x^5y^9z^{14}} )
- 3 divides 5 once with 2 remaining.
- ( = x \sqrt[3]{x^2} ).
- 3 divides 9 exactly three times.
- 3 divides 14 four times with 2 remaining.
- Result: ( xy^3z^4 \sqrt[3]{x^2z^2} ).
Rationalizing Denominators
Example: ( \frac{\sqrt{75x^7y^3z^{10}}}{8x^3y^9z^4} )
- Simplify by dividing exponents and simplifying square roots.
- Result: ( \frac{5x^2|z^3|\sqrt{3x^3}}{4|y^3|} ).
- Multiply by ( \sqrt{2} ) to rationalize.
- Final result: ( \frac{5x^2|z^3|\sqrt{6x^3}}{4|y^3|} ).
Final Example
Simplifying Complex Fractions
- Example: ( \sqrt[3]{\frac{16x^7y^4z^9}{54x^2y^9z^{15}}} )
- Simplify the cube roots and rationalize.
- Result: ( \frac{2x\sqrt[3]{x^2y}}{3y^2z^2} ).
Conclusion
- Mastered the technique of simplifying radicals with variables and exponents.
- Practice with different examples and rationalizations.
Thank you for watching and have a wonderful day!