Simplifying Radicals with Variables and Exponents
Simplifying Square Roots
- Example: Simplify ( \sqrt{x^5} )
- Index number is 2.
- Writing x five times: ( x \cdot x \cdot x \cdot x \cdot x ).
- Take out pairs of x: ( x^2 \sqrt{x} ).
- Alternative Method
- Dividing exponent by index: How many times does 2 go into 5?
- 2 goes into 5 two times with 1 remaining.
- Result: ( x^2 \sqrt{x} ).
More Examples
- ( \sqrt{x^7} )
- 2 goes into 7 three times with 1 remaining.
- Result: ( x^3 \sqrt{x} ).
- ( \sqrt{x^8} )
- 2 goes into 8 four times, no remainder.
- Result: ( x^4 ).
Simplifying Numbers with Square Roots
- Example: Simplify ( \sqrt{32} )
- Break into: 16 and 2 (since ( \sqrt{16} = 4 )).
- Result: ( 4 \sqrt{2} ).
- Example: ( \sqrt{50 x^3 y^{18}} )
- Simplify ( x^3 ): 2 goes into 3 one time with 1 remaining.
- Simplify ( y^{18} ): 2 goes into 18 nine times, no remainder.
- Consider absolute values for even index and odd exponent.
- Simplify ( \sqrt{50} ): Break into ( 25 \cdot 2 ).
- Combine: ( 5 \sqrt{2} x y^9 ).
Simplifying Cube Roots
- Example: ( \sqrt[3]{x^5 y^9 z^{14}} )
- Simplify ( x^5 ): 3 goes into 5 one time with 2 remaining.
- Simplify ( y^9 ): 3 goes into 9 three times, no remainder.
- Simplify ( z^{14} ): 3 goes into 14 four times with 2 remaining.
- Result: ( x y^3 z^4 \sqrt[3]{x^2 z^2} ).
Final Problem Examples
- Example: ( \sqrt[3]{16 x^{14} y^{15} z^{20} / 54 x^2 y^9 z^{15}} )
- Simplify 16 and 54 by dividing by 2: ( \sqrt[3]{8 / 27} = 2 / 3 ).
- Simplify exponents:
- ( x^{14-2} = x^{12} ).
- ( y^{15-9} = y^6 ).
- ( z^{20-15} = z^5 ).
- Combine: ( 2x^5 y^2 / 3y z^4 ).
- Result: ( 2 x \sqrt[3]{x^5 y^2 z^5} / 3 y z^4 ).
Key Takeaways
- To simplify square roots, remove pairs of variables based on the index.
- To simplify cube roots, divide exponents by 3 and handle remainders.
- Use absolute values for even indices that result in an odd exponent.
That's it for this lesson on simplifying radicals with variables and exponents. Thank you for watching!