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Simplifying Radical Expressions

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A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

  • how to simplify radicals using rational exponents
  • how to multiply radicals of the same root
  • how to multiply radicals of different roots

Simplifying Radicals using Rational Exponents

When simplifying roots that are either greater than four or have a term raised to a large number, we rewrite the problem using rational exponents. Remember that every root can be written as a fraction, with the denominator indicating the root's power. When simplifying radicals, since a power to a power multiplies the exponents, the problem is simplified by multiplying together all the exponents.
How to use rational exponents to simplify higher power and indexed radicals.
Using Rational Exponents to Simplify Radicals

Examples of Using Rational Exponents to Simplify Radicals
These problems look very complicated at first, but by just using the laws of exponents we can simplify very quickly.

Multiplying Radicals of the Same Root

When multiplying radical expressions of the same power, be careful to multiply together only the terms inside the roots and only the terms outside the roots; keep them separate. After multiplying the terms together, we rewrite the root separating perfect squares if possible. The rules of distributing and multiplying binomials (FOIL) apply to radicals as well.
How to multiply radicals of the same root together.
This video provides examples on how to multiply square roots and cube roots that contain variables.

How to handle multiplying square roots (including squaring radicals)
Multiplying Radicals

Multiplying Radical Expressions

Multiplying Radicals of Different Roots

To simplify two radicals with different roots, we first rewrite the roots as rational exponents. Before the terms can be multiplied together, we change the exponents so they have a common denominator. By doing this, the bases now have the same roots and their terms can be multiplied together. Next, we write the problem using root symbols and then simplify.
How to multiply radicals together when they are different roots.
Multiply Radicals with Different Indexes Using Rational Exponents - Different Radicand

Multiply and Divide Radicals with Different Indexes Using Rational Exponents - Same Radicand

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