On this page, we review important facts about roots and radicals.

Basic Learning Objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

  • Write a radical expression in lowest radical terms, and simplify by combining like radicals when possible.
  • Understand that when simplifying an even root of a power of a variable requires an absolute value, whereas this is not necessary for odd roots: \(\sqrt{x^2}=(x^2)^{1/2}={\color{red}|}x{\color{red}|}\), but \(\sqrt[3]{x^3}=(x^3)^{1/3}=x\).
  • Translate between radical notation and exponential notation (fractional exponents), and simplify radical expressions by using exponent laws.

Advanced Learning Objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

  • Multiply radicals, and expand a product of expressions involving terms with radicals using the distributive law.
  • Understand that in general, exponent laws only apply to non-negative bases, otherwise we would have absurd statements like \(-1 = (-1)^1 = ((-1)^2)^{\frac{1}{2}} = (1)^{\frac{1}{2}} = \sqrt{1} = 1\).
  • Simplify more complicated radical expressions.

To prepare for class

  • Watch the following video (by Heritagealgebra, 2m30s) which explains how to convert a root or radical into exponential form:

  • Watch the following video (by MyWhyU, 9m27s) which explains how to simplify radicals of integers and write them in lowest radical terms. It also explains how to simplify expressions of powers of variables, taking the absolute value for even roots:

  • Watch the following videos (by Tyler Wallace, 3m53s) which explain how to simplify radicals when there is a factor (“coefficient”) in front:

  • Watch the following video (by Emily Burns, 3m47s) which explains in detail when and why we need an absolute value when taking an even root of a power of a variable:

  • Watch the following videos (by Tyler Wallace, 5m39s) which shows some examples of simplifying radicals of powers of variables:

    • Watch the following video (by Tyler Wallace, 5m01s) which shows how to simplify a sum of several radicals, by combining like radicals after reducing each:

After class

  • Watch the following videos (by Tyler Wallace, each ~4m) which explain how to multiply and simplify expressions involving radicals:

  • Watch the following video (by Mathispower4u, 4m32s) which shows some more examples of simplifying radicals of variable expressions which require an absolute value:






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