Simplifying Radical Expressions Involving Multiplication and Division

Key Takeaways

• There are two types of multiplication when it comes to radicals: Multiplication inside the radical and multiplication of two radicals.
  • For multiplication inside radicals, we can use the factor grouping technique developed in the last lesson. This will work even with variables; just group powers of your variables with powers matching the radical.
  • For multiplication of two different radicals, if the power is the same on both radicals, the radicals can be combined into one big radical where you multiply all the stuff that was inside both radicals under the new big radical. This then gives you multiplication inside a radical that you can reduce according to the above bullet.
    • Ex: \(\sqrt{2x}\cdot \sqrt{3x}=\sqrt{6x^2}=\sqrt{6}\sqrt{x^2}=|x|\sqrt{6}\).

• If you have radicals with division inside, the big radical of the whole fraction can be turned into a radical of the numerator and of the denominator separately.
  • Ex: \(\sqrt{\frac{5x}{y^3}}=\frac{\sqrt{5x}}{\sqrt{y^3}}\)
• If you have a radical in the numerator and in the denominator with the same power on it, you can rewrite it as one big radical of a fraction (the opposite of the last bullet).
  • Ex: \(\frac{\sqrt[3]{x^2}}{\sqrt[3]{y^4}}=\sqrt[3]{\frac{x^2}{y^4}}\)

REMEMBER: When cancelling even powers with even roots on variables, you need to include absolute values around the variable outside the radical. For odd roots, no absolute values are necessary.