1. Simplifying Square Root Expressions

2. Numbers with a Root
Radical numbers are typically irrational numbers (unless they simplify to a rational number). Our calculator gives:
But the decimal will go on forever and not repeat because it is an irrational number. For the exact answer just use:
Some radicals can be simplified similar to simplifying a fraction.

3. Radical Product Property
ONLY when a≥0 and b≥0
For Example:
Equal

4. The square of whole numbers.
Perfect Squares
The square of whole numbers.
1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 ,
121, 144 , 169 , 196 , 225, etc

5. Simplifying Square Roots
Check if the square root is a whole number
Find the biggest perfect square (4, 9, 16, 25, 36, 49, 64) that divides the number in the root
Rewrite the number in the root as a product
Simplify by taking the square root of the perfect square and putting it outside the root
CHECK!
Note: A square root can not be simplified if there is no perfect square that divides it. Just leave it alone.
ex: √15 , √21, and √17

6. Simplifying Square Roots
Write the following as a radical (square root) in simplest form:
Simplify.
36 is the biggest perfect square that divides 72.
Rewrite the square root as a product of roots.
Ignore the 5 multiplication until the end.

7. Simplifying Square Roots
Simplify these radicals:

8. Adding and Subtracting Radicals
Simplify the expressions:
Always simplify a radical first.
Treat the square roots as variables, then combine like terms ONLY.

9. Multiplication and Radicals
Simplify the expression:
Use the Commutative Property to Rewrite the expression.
Simplify and use the Radical Product Property Backwards.
If possible, simplify more.
Conclusion: Multiply the numbers outside of the square root, then multiply the numbers inside of the square root. Then simplify.

10. Distribution and Radicals
Rewrite the expression:
3√ √3
Find the Sum.
15√36
-10√18
5√6
4√3 90
-30√2
12√18
-8√9
36√2
-24
Combine like terms.
Remember: Multiply the numbers outside of the square root, then multiply the numbers inside of the square root. Then simplify.

11. Fractions and Radicals
Simplify the expressions:
There is nothing to simplify because the square root is simplified and every term in the fraction can not be divided by 10.
Make sure to simplify the fraction.

12. Radical Quotient Property
ONLY when a≥0 and b≥0
For Example:
Equal

13. The Square Root of a Fraction
Write the following as a radical (square root) in simplest form:
Take the square root of the numerator and the denominator
Simplify.

14. Rationalizing a Denominator
The denominator of a fraction can not contain a radical. To rationalize the denominator (rewriting a fraction so the bottom is a rational number) multiply by the same radical.
Simplify the following expressions:

15. WARNING
In general:
For Example:
Not Equal