Lecture Notes: Simplifying Radical Expressions

Objective

• To simplify radical expressions using the laws of radicals.

Introduction to Radicals and Perfect Squares

• A perfect square can be factored to simplify a radical.
  • Example: √8 can be simplified as 2√2 because 8 is 4×2.

Simplifying Common Radicals

• √12 = 2√3
• √18 = 3√2
• √20 = 2√5
• √24 = 2√6
• √27 = 3√3
• √28 = 2√7
• √32 = 4√2
• √40 = 2√10
• √44 = 2√11
• √45 = 3√5
• √48 = 4√3
• √50 = 5√2
• √52 = 2√13
• √54 = 3√6
• √56 = 2√14
• √60 = 2√15
• √63 = 3√7
• √68 = 2√17
• √72 = 6√2
• √75 = 5√3
• √76 = 2√19
• √80 = 4√5
• √84 = 2√21
• √88 = 4√2 or 2√22
• √92 = 2√23
• √96 = 4√6
• √98 = 7√2
• √99 = 3√11

Laws of Exponents in Simplifying Radicals

• The square root of a product of two numbers is the product of their square roots.
• Example: Simplifying √(8x^5y^6z^13), factorize into perfect squares:
  • √4 * √(x^4) * √(z^12) simplifies to 2x^2y^3z^6√(2xz).

Reducing Index to Lowest Possible Order

• Convert the nth root of a number into an exponential form for simplification.
  • Example: Simplify 20th root of 32m^15n^5.
  • Express as (2^5 * m^15)^(1/20) = 2^(1/4) * m^(3/4).
  • Result: 4th root of 2m^3n.

Examples and Simplifications

1. √(x^5):
   • Factor as √(x^4 * x) = x^2√x.
2. Cube root of b^7:
   • Factor as ∛(b^6 * b) = b^2∛b.
3. √(63m^2):
   • Factor as √(9 * 7 * m^2) = 3m√7.
4. √(60x^4y^5):
   • Factor as √(15 * 4 * x^4 * y^4 * y) = 2x^2y^2√(15y).

Using Cube Roots and Higher Indices

• Cube root of 125x^5y^18:
  • Result: 5xy^6∛(x^2).
• Cube root of 56m^10n^12:
  • Result: 2m^3n^4∛(2m).

Combining and Simplifying Radicals with Coefficients

• Negative coefficients and mixed indices:
  • Example: Cube root of -16x^6y^3z^7.
  • Simplified result: -2xyz^3∛(2yz).

Examples of Fourth Roots and Higher

• Fourth root of a term:
  • Example: 81x^24y^36, factor as (81 * x^24 * y^36)^1/4, simplified to 3x^6y^9.

Advanced Simplifications

• Combining multiple radical expressions:
  • Example: Cube root of (2^4 * 5^5), simplification to 10∛50.*

Conclusion

• Techniques for simplifying radical expressions are critical for solving complex mathematical problems.
• Practice and familiarity with perfect squares and indices enhance simplification skills.
• Understanding the laws of exponents is essential for mastering radical expressions.