Operations-on-Radicals

Page 1

ADDITION AND SUBTRACTIONS OF RADICALS

  • Introduces the topic of adding and subtracting radical expressions.

Page 3

Review: Similar or Dissimilar?

  • Task: Determine if given sets of radicals are similar or dissimilar.

  • Similar: Groups marked as similar.

  • Dissimilar: Groups marked as dissimilar.

Page 4

Similar Radicals

  • Definition: Similar radicals have the same index and the same radicand.

  • Addition and subtraction operations:

    • Addition: ( x\sqrt{a} + y\sqrt{a} = (x + y)\sqrt{a} )

    • Subtraction: ( x\sqrt{a} - y\sqrt{a} = (x - y)\sqrt{a} )

Page 5

Instructions for Operations

  • Perform the indicated operation on the radicals:

    1. ( \frac{4}{2} + 3\sqrt{2} )

    2. ( \frac{10}{3} - \frac{15}{3} )

    3. ( 5\sqrt{7} - 33 + 257 )

    4. ( 12\sqrt{6} - 10\sqrt{6} + \sqrt{6} )

    5. ( 2x\sqrt{5} + 5x\sqrt{5} - 10x\sqrt{5} )

    6. ( \frac{3}{11} + 5\sqrt{11} + 2\sqrt{11} )

Page 6

Examples – Performing Operations and Simplifying Radicals

  1. ( 6\sqrt{2} + 9\sqrt{2} - 4\sqrt{2} = (6 + 9 - 4)\sqrt{2} = 11\sqrt{2} )

  2. ( 9\sqrt{5} - 4\sqrt{5} - 6\sqrt{3} + 2\sqrt{3} = (9 - 4)\sqrt{5} + (-6 + 2)\sqrt{3} = 5\sqrt{5} - 4\sqrt{3} )

  3. ( \frac{7}{27} + 3\sqrt{75} )

  4. ( 5\sqrt{32} - 3\sqrt{108} )

  5. ( \frac{5}{8} - 3\sqrt{50} + \frac{8}{72} )

  6. ( 2\sqrt{24a^3b^4} + 3a\sqrt{81b^4} )

Page 7

Multiplication of Radical Expressions

  • When multiplying radicals, consider: A. Indices are the same:

    • Use product rule for radicals ( n\sqrt{a} \cdot n\sqrt{b} = n\sqrt{ab} )

    • Simplify resulting radicand if needed.

Page 9

Multiplying Radicals with Different Indices but Same Radicands

  1. Steps:

    • Transform radicals into fractional exponents.

    • Apply exponent laws: ( a^m \cdot a^n = a^{m+n} )

    • Rewrite the product as a single radical.

    • Simplify if necessary.

Page 11

Radicals with Different Indices and Radicands

  1. Steps:

    • Transform to fractional exponents.

    • Change to similar fractions.

    • Rewrite as a single radical.

    • Simplify if necessary.

Page 13

Multiplying a Radical by a Binomial

  • Use distributive property to expand.

Page 17

Multiplying a Binomial Radical Expression

  • Similar to the FOIL method:

    • Multiply each term in the first binomial by each term in the second.

Page 19

Conjugates

  • Conjugate of a binomial changes the sign of the second term.

  • Multiplication results in rational numbers.

  • Product of conjugates is a difference of squares: ( a^2 - b^2 ).

Page 20

Examples Using Conjugates

  1. ( (\sqrt{3} + \sqrt{7})(\sqrt{3} - \sqrt{7}) )

    • Use difference of squares property to simplify: ( 3 - 7 = -4 ).

Page 22

Instructions for Calculating Products and Conjugates

  • Tasks include multiplying radical expressions and finding conjugates:

    • Examples provided for practice and skills development.

Page 23

Division of Radical Expressions

  • Rewrite radical divisions to fractional form and rationalize.

  • Rationalization: Simplifying to remove radicals from the denominator.

Page 24

Examples of Division

  1. ( \frac{V72}{V6} )

    • Factor and simplify the radicand.

Page 27

Rationalizing Binomial Denominators

  • Multiply by conjugate to remove radicals from denominators.

  • Apply difference of squares concept for simplifying.

Page 29

Final Example on Rationalization

  1. Example of rationalization using conjugates.

    • Steps include multiplying by conjugate and simplifying the result.