Simplifying Radicals with Variables and Exponents

Key Concepts

Radicals: Expressions that involve roots, such as square roots and cube roots.
Index Number: The degree of the root (e.g., square root has index 2).
Simplifying Radicals: The process of reducing radicals by removing perfect squares or cubes.

Example 1: ( \sqrt{x^5} )
- Write out (x) five times: ( x \cdot x \cdot x \cdot x \cdot x )
- Take out pairs: ( \sqrt{x^5} = ext{two pairs + one remaining} \rightarrow x^2 \sqrt{x} )
How to Calculate:
- Divide the exponent by the index:
  - (5 \div 2 = 2 \text{ (2 pairs with 1 remaining)})
Example 2: ( \sqrt{x^7} )
- (7 \div 2 = 3 \text{ (3 pairs with 1 remaining)})
Example 3: ( \sqrt{x^8} )
- (8 \div 2 = 4 \text{ (4 pairs with no remaining)})
Example 4: ( \sqrt{x^9} )
- (9 \div 2 = 4 \text{ (4 pairs with 1 remaining)})

Example: ( \sqrt{32} )
- Break down into perfect squares: (32 = 16 \times 2)
- ( \sqrt{32} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} )
Example: ( \sqrt{50x^3y^{18}} )
- Break down: (50 = 25 \times 2)
- Powers: (x^3 = x^2 ext{ (1 pair, 1 remaining)}), (y^{18} = y^9 ext{ (9 pairs)})
- Result: (5\sqrt{2}x\sqrt{y^0} = 5\sqrt{2}x)

Example: ( \sqrt[3]{x^5y^9z^{14}} )
- Calculate pairs:
  - (5 \div 3 = 1 \text{ (1 pair with 2 remaining)})
  - (9 \div 3 = 3 \text{ (3 pairs with no remaining)})
  - (14 \div 3 = 4 \text{ (4 pairs with 2 remaining)})

Example: ( \frac{\sqrt{75x^7y^3z^{10}}}{8x^3y^9z^4} )
- Simplify numerator and denominator:
  - (75 = 25 \times 3), (8 = 4 \times 2)
  - Subtract exponents in division: (7-3=4), (3-9=-6), (10-4=6)
- Result after simplification:
  - Radicals: (5\sqrt{3}) and (2)
  - Absolute values needed due to even index and odd exponents.

Simplifying radicals involves breaking down the numbers and variables, taking out pairs based on the index, and ensuring calculations are correct with remainders.
Always be aware of absolute values when dealing with even indices and odd exponents.