MODULE 7

• DISCIPLINE • GOOD TASTE • EXCELLENCE
• Writer: Gemma Y. Bandoquillo
• Cover Illustrator: Joel J. Estudillo
• MATHEMATICS Quarter 2: Module 7 Operations on Radical Expressions
• Department of Education National Capital Region SCHOOLS DIVISION OFFICE MARIKINA CITY
• Radical expressions


  Expressions containing square roots
• In this module you will learn how to perform operations on radical expressions: addition, subtraction, multiplication, division
• Operations on radical expressions

  • Addition
  • Subtraction
  • Multiplication
  • Division
• Radical expressions


  • They can be simplified by combining like radicals (same radicand and index)
  • Radicals should be simplified before performing operations
• Like radicals


  Radicals with the same radicand and index
• Simplifying like radicals


  1. Combine by adding/subtracting coefficients and keeping common radical

  2. Simplify radicals first if not in simplest form
• Only like radicals can be combined by adding or subtracting
• Multiplying radicals


  1. Multiply the radicands

  2. Factor the radicand to get the greatest perfect square factor

  3. Simplify by removing perfect nth powers from the radicand
• Multiplying binomials involving radicals


  Use the property for the product of two binomials: (a ± b)(c ± d) = ac ± ad ± bc ± bd<|>Simplify by removing perfect nth power from the radicand or by combining similar radicals
• Multiplying radicals of different orders

  1. Transform the radicals into fractional exponents

  2. Change the fractional exponents into similar fractions

  3. Transform to radical form and multiply

  4. Simplify
• To multiply radicals with the same indices, use the Product Property of Square Root: √a^n * √b^n = √(a*b)^n
• To multiply radical expressions with variables, use the distributive property: √x(√m + a) = √x√m + √xa
• To multiply binomials that contain radicals, use the property for the product of two binomials: (√m + a)(√m - a) = m - a^2
• To multiply the square of binomials that contain radicals: (√a + y)^2 = a + 2√ay + y^2
• To multiply radicals of different orders:

  1. Transform the radicals into fractional exponents

  2. Change the fractional exponents into similar fractions

  3. Transform to radical form and multiply

  4. Simplify
• Algebraic expressions with variables


  We use radicals
• √�� (√𝑚 + 𝑎)


  = √(𝑥(√𝑚 + 𝑎))
• Multiplying binomials that contain radicals

  Use the distributive property
• (√�� + 𝑎) (√𝑚 − 𝑎)


  = 𝑚 - 𝑎^2
• (√𝑎 + 𝑦)^2

  = 𝑎 + 2√��𝑦 + 𝑦^2
• Multiplying radicals of different orders

  1. Transform the radicals into fractional exponents

  2. Change the fractional exponents into similar fractions

  3. Transform to radical form and divide
• y > 0

  1. √5xy2

  2. (5xy2)1/2

  3. √27x3y6

  4. (27x3y6)1/3

  5. 3
• Divide √2m5n3 / √3m2n2 4, where m > 0 and n > 0

  √(125x3y6 / 729x6y12) 6
• √2m5n3 / √3m2n2 4


  m2n√(108n) / 3