Multiplication and Division of Square Roots

1. Multiplication of Square Roots : Multiply the numbers inside the radical with each other, and multiply the numbers outside the radical with each other.
   For and , and and as rational numbers:


2. Division of Square Roots : Divide the numbers inside the radical with each other, and divide the numbers outside the radical with each other.
   For and and as rational numbers:

Manipulating Expressions with Radicals

1. If the number inside the radical is a perfect square, you can simplify the expression by taking it out of the radical.
   For and :
2. A positive number outside the radical can be written inside the radical.
   For and :

Rationalizing the Denominator

1. Rationalizing the Denominator : When the denominator of a fraction contains a radical, multiply both the numerator and denominator by the same non-zero value to make the denominator a rational number.
   
   
2. Methods for Rationalizing the Denominator
   For and being rational numbers:
3. (provided )
4. (provided )


3. If the denominator contains a perfect square inside the radical, simplify it using before rationalizing the denominator.
   (Example)
   
   
4. Mixed Calculations Involving Multiplication and Division of Square Roots
1. Rewrite division as multiplication by the reciprocal.
2. Simplify the expression using properties of square roots.
3. If the denominator contains an irrational number, rationalize the denominator.