Algebra 1 - Multiplying and Dividing Radical Expressions


  • In multiplying radical expressions, we use the Product Rule: If x and y represents nonnegative real numbers, then x·y=xy.
  • For higher indices, we have xn·yn=xyn where n>1, x and y are nonnegative real numbers. 
  • In dividing radical expressions, the Quotient Rule holds: Given that x and y where y0 and n is an integer greater than 1, then xnyn=xyn. If n is an even integer, then both x and y are nonnegatives. 


Important Key Points:

  • We can multiply radical expressions having the same index yet different radicands. In some cases, after multiplying, we need to simplify the result.
  • We can also multiply two or more radicals with different indices by following a series of steps. 
  • Certain algebraic rules in multiplying polynomials (i.e. special product cases) can apply to the multiplication of radical expressions.
  • In the division of radicals with two terms in the denominator, we can rationalize this type of radical expression by multiplying the numerator and denominator by the conjugate of the denominator.


Solved Examples

Example 1. Find the product: 7xy2x·4z38x3


7xy2x·4z38x3=28xyz3 16x4=28xyz3·4x2=112x3yz3


Example 2. Find the product: 65ab33·725a4b63


65ab33·725a4b63=42 125a5b63=42 125a3·a2·b63=210ab2 a23


Example 3. What is the product of 7 and 53?




Example 4. Find the product: 7x-3y7x+3y




Example 5. Give the product: 5a-b7c2




Example 6. Find the quotient: 1728+314-2987




Example 7. Simplify the expression 73+773-7


73+773-7·73+773+7=73+72732-72=154+1421140 =1411+2114·10=11+2110

Cheat Sheet

  • To simplify the expression x-yx+y, Quotient Rule is applicable where x+y>0 and x-y0.
  • The conjugate of the denominator of 11-3x27x+36 is 27x-36.
  • To simplify the expression 4x3+y, rationalize the denominator by multiplying the denominator by its conjugate which is x23-y x3+y2. This is based on the concept of x+yx2-xy+y2=x3+y3.
  • When dividing two radicals of the same order, use the Quotient Rule and siimplfy it by rationalizing the denominator.
  • In multiplying two finite nested radicals like 7-36·7+36, we treate 7-36 and 7+36 as radicands.
  • To rationalize 7x5, multiply both numerator and denominator by x45.

Blunder Areas

  • The process of multiplying the square root of two negative numbers such as -16·-9=144 is incorrect since the radicand must be nonnegative real numbers. -16=4i and -9=3i.
  • The square root of negative numbers is not defined under the real number system.
  • When multiplying and dividing radical expressions, it is imperative to simplify the final result whenever possible.