When dividing with square roots, we must treat coefficients, numbers within our base, and variables within our base all differently. Lets begin with a simple problem involving only numbers in our bases.

Example:

Instead of using two separate square roots, make this one square root with division occurring inside it.


Now we will examine a problem with coefficients.

Example :

Simplify the division of the coefficients, while also combining the bases under one radical.


So now we have: 2√8

Be careful! We are not done yet. Examine the base. The perfect square 4 can be factored out. Therefore:


Finally, we will examine a problem that also has variables within the bases. When this happens, we must remember the rules about dividing with powers.


Notice that, since we have division with exponents, we subtract with our exponents. Putting all of this together, we get:


Practice Problems

Simplify.

1. √240 / √10

2. 4√400 / 2√8

3. -10√16x³ / 5√2x

Answers

1. √240 / √10

√( 240 / 10 )

√24

√( 4 x 6 )

2√6


2. 4√400 / 2√8

(4 / 2)√( 400 / 8 )

2√50

2√( 25 x 2 )

(2 x 5)√2

10√2


3. -10√16x³ / 5√2x

(-10 / 5) = -2 - Coefficients

√( 16 / 2 ) = √8 = √( 4 x 2 ) = 2√2 - Numbers in base

√(x³ / x) = √x^{3 - 1} = √x² - Variables in base

All together: (-2)(2√2)(√ x² )

( -2 x 2 )√ (2x² )

-4√2x²

Related Links: Math algebra Adding Square Roots Introduction to Cube Roots Simplifying Square Roots Multiplying Square Roots with Exponents Multiplying Square Roots Worksheets Dividing Square Roots Worksheets