1. 11-1: Simplifying Radicals
Essential question: How do you simplify a radical?

2. 11-1: Simplifying Radicals
Radical expressions like contain a radical.
You read as “the square root of the quantity x plus 3”
You can simplify radicals by removing perfect-square factors from underneath the radical (called the “radicand”).
Multiplication Property of Square Roots
For every number a > 0 and b > 0,
Example:

3. 11-1: Simplifying Radicals
Example 1: Simplify
Method A (Finding Perfect Squares)
Perfect squares are numbers that can be produced by multiplying a number by itself (e.g. 25 is a perfect square since 25 = 5 ● 5)
Find any perfect squares that can divide the radicand, and rewrite the radicand as the product of two radicals
Since 192 = 64 ● 3,

4. 11-1: Simplifying Radicals
Example 1: Simplify
Method B (Using Factor Trees)
Break down the number under the radical into its prime factorization (like we did with finding a GCF)
If there are any pairs, they merge together for one to come outside the radical
Numbers brought outside the radical get multiplied together, as well as numbers left inside the radical

5. 11-1: Simplifying Radicals
Your Turn
Simplify each radical

6. 11-1: Simplifying Radicals
You can simplify radical expressions that contain variables.
A variable with an even exponent is a perfect square (e.g. x8 = x4 ● x4)
A variable with an odd exponent (other than 1 or -1) is the product of a perfect square and the variable (e.g. x7 = x3 ● x3 ● x)
You can also break variable down (like we did with GCFs) and remove pairs as a singular entity
In Algebra 1, we will assume that all variables under radicands represent positive numbers

7. 11-1: Simplifying Radicals
Example 2: Simplify
45 = 3 ● 3 ● 5
a5 = a2 ● a2 ● a
YOUR TURN
Simplify each radical

8. 11-1: Simplifying Radicals
You can use the multiplication property to write Often times, it’s easier to multiply two radicals together and then simplify.
Example 3: Multiplying Two Radicals

9. 11-1: Simplifying Radicals
Your Turn
Simplify each radical

10. 11-1: Simplifying Radicals
Example 4: Real-World Application
You can use the formula to estimate the distance d in miles to a horizon when h is the height of the viewer’s eyes above the ground in feet.
Suppose you are looking out on a second floor window 25 feet above the ground. Find the distance you can see to the horizon. Round your answer to the nearest mile.
You can see about 6 miles.

11. 11-1: Simplifying Radicals
Your Turn
Suppose you are looking out a fourth floor window 52 ft above the ground. Use the formula to estimate the distance you can see to the horizon. Round your answer to the nearest mile.
9 miles

12. 11-1: Simplifying Radicals
Division Property of Square Roots
For every number a > 0 and b > 0,
Example:

13. 11-1: Simplifying Radicals
When the denominator is a perfect square, it’s easier to simplify separately
Example 5: Simplify each expression
A) B)

14. 11-1: Simplifying Radicals
Your Turn
Simplify each expression
A) B) C)

15. 11-1: Simplifying Radicals
When the denominator is not a perfect square, it may be easier to divide first and then simplify
Example 6: Simplify each expression
A) B)

16. 11-1: Simplifying Radicals
Your Turn
Simplify each expression
A) B) C)

17. 11-1: Simplifying Radicals
Sometimes, the denominator may not come out to be a perfect square. Fractions are in their simplest form when there are no roots in the denominator, which means you may have to rationalize a denominator.
To rationalize, you multiply the numerator AND denominator by the whatever radical expression is on the denominator of the fraction.

18. 11-1: Simplifying Radicals
Example 7a: Rationalizing a Denominator
Multiply by to make a perfect square
Multiply square roots
Simplify

19. 11-1: Simplifying Radicals
Example 7b: Rationalizing a Denominator
Simplify the denominator first
Multiply by
Multiply numerator & denominator
Simplify

20. 11-1: Simplifying Radicals
Your Turn
Simplify by rationalizing the denominator
A) B) C)

21. 11-1: Simplifying Radicals
Assignment
Worksheet #11-1
1 – 81 (odds)
Due 5/26 (next Tuesday)