10-3: Estimating Radicals (Square Roots)
11-1: Simplifying Radicals (Square Roots)
Objectives:
• You will be able to estimate the value of a radical expression (square root) to get an approximate value.
• You will be able to simplify radical expressions (square roots) to get an exact value.
• You will be able to simplify radical expressions that use multiplication.
• You will be able to simplify radical expressions that use division.
Perfect Squares:
When we are finding a square root, we are really working backwards from the area of a square to find the length of one side of that square.
Area = 1
Side = 1
Area = 4
Area = 9
Side = 2
Side = 3
Area = 16
Side = 4
16
25
36
49
64
81
100
121
144
ESTIMATING SQUARE ROOTS:
: What two perfect squares is 15 between? Which perfect square is 15 closer to? ____________________
__________
Estimate the value of (as a decimal) based on your previous answer. __________
Practice Problems:
1.
2.
3.
4.
Review:
Make factor trees for the following numbers.
5.
6.
7.
8.
SIMPLIFYING SQUARE ROOTS:
9.
10.
11.
12.
THE DIFFERENCE BETWEEN ESTIMATING SQUARE ROOTS AND SIMPLIFYING SQUARE ROOTS IS…
Estimating uses ____________________ while simplifying does not.
Simplifying uses the __________________ whenever appropriate while estimating does not. Practice Problems:
Estimate and simplify each radical expression (show factors for simplifying):
13.
estimate: ______________
simplified: ______________
14.
estimate: ______________
simplified: ______________
15.
estimate: ______________
simplified: ______________
MULTIPLYING RADICAL EXPRESSIONS:
When you multiply two radical expressions, multiply the __________________ of one with the __________________ of the other and multiply the ________________ of one with the ________________ of the other. Then simplify like before.
Examples & Practice Problems:
Multiply and simplify each radical expression.
16.
17.
18.
19. You can use the formula to estimate the distance in miles to a horizon when h is the height of the viewer’s eyes above the ground in feet. Estimate the distance a visitor at the Washington Monument (height = 500 feet) can see to the horizon from the observation windows. Round your answer to the nearest mile.
DIVIDING RADICAL EXPRESSIONS:
When dividing radical expressions,
“simplified” means that there are no square roots in the _________________.
When dividing radical expressions, three things could happen:
Case #1: The denominator could be a ________________ ______________,
Case #2: The numerator and denominator could ______________ evenly, or
Case #3: The denominator does not work out nicely and has to be ________________________.
Case #1: When the denominator is a perfect square:
Separate the radical expression into two parts (top and bottom) and deal with each part separately. Simplify the numerator then if necessary simplify top to bottom.
20.
21.
22.
Case #2: When the numerator and denominator divide evenly:
Divide the numerator and denominator then simplify like we did earlier.
23.
24.
25.
Case #3: When the denominator is not a perfect square and does not divide evenly with the numerator.
The denominator needs to be “rationalized”…
we need to make it a perfect square.
26.
27.
28.