Math 101 Intermediate Algebra
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Multiplying and Simplifying Radicals
Chapter 8, Section 3


Remember that for the rest of this chapter, it will be assumed that all variables represent non-negative real numbers. So, for example

 =  a, a    0.


Product Rule For Radicals
For non-negative real numbers a and b,

which holds since


Simplifying Radicals Whose Radicands Are Natural Numbers

  1. Write the radicand as the product of two numbers, one of which is the largest perfect power number for the given index.
  2. Use the product rule to write the expression as a product of roots.
  3. Find the roots of any perfect power numbers. This means, simplify.
  4. You are done when the radicand has no factor that is a perfect power.

You obtain the numbers that are perfect powers by squaring the natural numbers.


Simplifying Radicals Whose Radicands Are Variables

  1. Write the radicand as the product of two numbers, one of which is the largest perfect power of the variable for the given index.
  2. Use the product rule to write the expression as a product of radicals. Place all perfect powers under the same radical.
  3. Find the roots of any perfect powers. This means, simplify.
  4. You are done when the radicand has no more perfect powers of any variable for the given index.


Simplifying Radicals

  1. If the radicand contatins a numerical factor, write it as a product of two numbers, one of which is the largest perfect power number for the given index.
  2. Write the radicand as the product of two numbers, one of which is the largest perfect power of the variable for the given index.
  3. Use the product rule to write the expression as a product of radicals. Place all perfect powers (numbers and variables) under the same radical.
  4. Find the roots of any perfect powers. This means, simplify.
  5. You are done when the radicand has no factor of the number that is a perfect power and no more perfect powers of any variable for the given index.


Multiplying Radicals

  1. Multiply radicals using the product rule.
  2. After multiplying, use the steps to simplify radicals stated above.