Conjugates / Dividing by Square Roots (page 4 of 7)

Sections: Square roots, More simplification / Multiplication, Adding (and subtracting) square roots, Conjugates / Dividing by square roots, Rationalizing denominators, Higher-Index Roots, A special case of rationalizing / Radicals & exponents / Radicals & domains

• Simplify

I do the multiplication:

Then I complete the calculations by simplifying:

• Simplify:

I do the multiplication:

Then I simplify:

Given the radical expression , the "conjugate" is the expression .

The conjugate (KAHN-juh-ghitt) has the same numbers but the opposite sign in the middle. So not only is the conjugate of , but is the conjugate of .

When you multiply conjugates, you are doing something similar to what happens with a difference of squares:

When you multiply the factors a + b and ab, the middle "ab" terms cancel out:

The same thing happens when you multiply conjugates:

We will see shortly why this matters. To get to that point, let's take a look at fractions containing radicals in their denominators.

Dividing by Square Roots

Just as you can swap between the multiplication of radicals and a radical containing a multiplication, so also you can swap between the division of roots and one root containing a division.

• Simplify:
• I can simplify this by working inside, and then taking the square root:

...or else by splitting the division into two radicals, simplifying, and cancelling:

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 Cite this article as: Stapel, Elizabeth. "Conjugates / Dividing by Square Roots." Purplemath. Available from     http://www.purplemath.com/modules/radicals4.htm. Accessed [Date] [Month] 2016

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