Radical Simplification Lecture Notes

Introduction

• Third video in a series for quarantine students.
• Focus on dealing with variables and numbers under square roots.
• Previous videos cover square roots and cube roots.

Key Concepts

Reducing Radicals with Variables

• Treat numbers and variables separately when simplifying radicals.

Dealing with Square Roots

1. Example: (\sqrt{50x^7})

   • Perfect Squares: List includes 1, 4, 9, 16, 25, 36, 49, etc.
   • (50) is not a perfect square.
     • Find largest perfect square that divides 50: (25).
     • (\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}).
   • Variables: Check if the exponent is divisible by 2.
     • (x^7) is not perfect (7 not divisible by 2).
     • Break into (x^6) and (x^1).
     • (\sqrt{x^7} = x^3\sqrt{x}).
   • Final Simplified Form: (5x^3\sqrt{2x}).

2. Example: (\sqrt{12x^9})

   • Find perfect square that divides 12: (4).
   • (\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}).
   • (x^9) is not perfect; break to (x^8) and (x^1).
   • (\sqrt{x^9} = x^4\sqrt{x}).
   • Final Form: (2x^4\sqrt{3x}).

3. Example: (\sqrt{45x^{12}y^{17}})

   • Find perfect square for 45: (9).
   • (\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}).
   • (x^{12}) is perfect: (x^6).
   • (y^{17}) is not perfect; break to (y^{16}) and (y^1).
   • (\sqrt{y^{17}} = y^8\sqrt{y}).
   • Final Form: (3x^6y^8\sqrt{5y}).

Conclusion

• Emphasizes understanding the process of simplification, irrespective of the method used.
• Next video will address cube roots.

Study Tips

• Recall lists of perfect squares for quick reference.
• Practice breaking down exponents on variables to simplify roots.
• Utilize online resources and videos for additional support during quarantine learning.