Answer:
(-3√2 + 2√10) / 2√2
Explanation:
1. Simplify 9 / -6:
- Find the greatest common divisor (GCD) of 9 and 6: GCD(9,6) = 3
- Divide both numerator and denominator by 3:
9 ÷ 3 = 3
-6 ÷ 3 = -2
- Result: 9 / -6 = -3/2
2. Rewrite the expression:
- Original: 9 / -6 + √5
- After simplification: -3/2 + √5
- Note: √5 can be written as √5/1 to prepare for common denominator
3. Get a common denominator:
- Current fractions: -3/2 and √5/1
- Least common multiple of denominators: LCM(2,1) = 2
- Multiply √5/1 by 2/2 (which equals 1) to get equivalent fraction with denominator 2:
√5/1 * 2/2 = 2√5/2
- Now we have: -3/2 + 2√5/2
- Add numerators: (-3 + 2√5) / 2
4. Rationalize the denominator:
- Current expression: (-3 + 2√5) / 2
- Multiply both numerator and denominator by √2:
(-3 + 2√5) / 2 * √2/√2
- This is valid because √2/√2 = 1, so we're multiplying by 1
- Simplify: ((-3 + 2√5) * √2) / (2 * √2) = ((-3 + 2√5) * √2) / 2√2
5. Simplify the numerator:
- Distribute √2: (-3√2 + 2√5√2) / 2√2
- Simplify √5√2:
√5 * √2 = √(5*2) = √10
- Final result: (-3√2 + 2√10) / 2√2
