Multiplication of Radicals

• How to multiply radicals

  1. Apply the product rule for radicals

  2. Multiply numerical coefficients

  3. Multiply radicands

  4. Simplify if necessary
• Product rule for radicals

  The nth root of a times the nth root of b is equal to the nth root of a*b
• Multiplying radicals with the same indices

  • Square root of 12 times square root of 3 = square root of 36 = 6
  • Cube root of 3 times cube root of 9 = cube root of 27 = 3
• Multiplying radicals with different indices but the same radicand

  • Square root of y times square root of y^5 = square root of y^6 = y^3
  • Fourth root of 4y^3 times fourth root of 12y^2 = fourth root of 48y^5 = 2y
• Multiplying radicals with different indices and different radicands

  • Square root of 5x times square root of 15 = square root of 75x = 5√3x
  • Three square root of 5 times two square root of 6 = six square root of 30
• Multiplying radicals with different indices

  Transform the radicals to powers with rational exponents<|>Add the exponents<|>Rewrite the product as a single radical<|>Simplify the resulting radical if necessary
• Multiplying radicals with different indices and the same radicand

  • Square root of 5 times cube root of 5 = (5^(1/2))*(5^(1/3)) = 5^(1/2 + 1/3) = 5^(5/6) = (6th root of 5)^5
• Multiplying radicals with different indices and different radicands

  • Cube root of 2 times square root of 3 = (2^(1/3))*(3^(1/2)) = (2*3)^(1/12) = (6th root of 6)
• Multiplying polynomial radical expressions by monomials

  Use the distributive property
• l nothing so 4 times 27 the answer is six root of 108 okay another example maritime cube root of two times fourth root of four so it transformed that n into rational exponent so that is two raised to one third times four raised to one fourth and then going nothing similar fraction and one third so one fourth and okay uh in denominator not ten yes so twelve divide three so that is four times one k four accumulator detail twelve divide four three times one kayak three a numerator detox similar fraction so going index so 12 root of 2 raised to 4 times 12 root of 4 cubed and 2 to the fourth power is 16 let's say 2 times 2 4 times 2 8 times 2 16 4 cubed that is 64 bucket 4 times 4 16 times 4 64. and then okay back at sixth okay nothing and 12 okay so therefore since same young indexing laptop the answer is 12 root of one thousand twenty four
• Multiply polynomial radical expression by monomials

  Apply distributive property
• The answer is square root of 6 plus 2 square root of 10
• Multiply square root of x times 2 square root of x

  1. Square root of x times 2 square root of x

  2. Square root of x times -3 square root of 2
• The answer is 2x - 3 square root of 2x
• The answer is 8 square root of 60 - 6 square root of 36 + 4 square root of 2
• Multiply polynomial radical expressions using the FOIL method

  1. First term

  2. Outer term

  3. Inner term

  4. Last term
• The answer is 1 - square root of 3
• The answer is 12 + 3 square root of 6 - 3 square root of 2 - 4 square root of 3
• Square (square root of 2 - 5)


  1. Square of square root of 2

  2. -2 * square root of 2 * square root of 5

  3. -5 * -5
• The answer is 27 - 10 square root of 2
• The answer is 27 + 36 square root of 2