Addition and Subtraction of Square Roots

Addition and subtraction of square roots are done by combining terms with the same numbers under the square root.
If and are rational numbers and is an irrational number:

Notes

• •If the number under the square root is in the form of , simplify it as to make further calculations.
  
• •If the denominator contains an irrational square root, rationalize it before performing further operations.
  
• •If the numbers under the square roots are not the same, they cannot be further simplified.
  

Mixed Calculations with Square Roots

1. Distributive Property with Square Roots
   For :


2. Rationalizing the Denominator using Distributive Property
   For :
   
   
   
3. Steps for Mixed Calculations with Square Roots
1. Use the distributive property to simplify expressions with parentheses.
2. If the number under the square root has a square factor, take it outside the square root.
3. If the denominator contains an irrational square root, rationalize it.
4. Perform multiplication and division first, then addition and subtraction.

Integer and Decimal Parts of Irrational Numbers

1. Integer and Decimal Parts of Irrational Numbers
1. An irrational number can be divided into an integer part and a decimal part.
   (Irrational number) (Integer part) (Decimal part) (Decimal part)
2. The decimal part of an irrational number is the difference between the irrational number and its integer part.
   If and is an integer such that,
   


2. Using Subtraction to Compare Real Numbers
   The comparison between two real numbers and is determined by the sign of :
1. If , then
2. If , then
3. If , then

Theorem on Equality of Irrational Numbers

1. If and are rational numbers and is an irrational number, then if
   , both and .
2. Explanation:
   If and , then , which contradicts the fact that is irrational.
   Therefore .
   Since and ,
   
   
3. If , , , and are rational numbers and is irrational, then if , it follows that and .
4. Explanation:
   Rearranging gives .
   By the previous result, and , so and .