NOTES: SIMPLIFYING SQUARE ROOTS
Simplify the Square Roots.
Step 1.
Step 2.
Step 3.
1.
50
2.
200
3.
48
4.
128
5.
2 54
5.
2 48
DAY 1
NOTES: SIMPLIFYING CUBE ROOTS
Step 1.
Step 2.
Step 3.
Simplify the Cube Roots.
3.
3
16
6.
3.
5 3 64
DAY 1
NOTES: SIMPLIFYING NTH ROOTS
DAY 2
Step 1.
Step 2.
Step 3.
Simplify the Cube Roots.
1.
34 256
3.
3 4 32 5.
3 (x
+ 4)3 2.
7
x7
4.
10 5 32 6.
7
x7 NOTES: MULTIPLY AND DIVIDE RADICALS
Step 1.
Step 2.
Step 3.
Radical Operations: Multiply, Divide
11.
8 i
13.
75
3
17.
2
12.
16.
18.
33 4 i 5 3 2
3 i
5
DAY 3
NOTES: RATIONALIZING THE DENOMINATOR
Textbook Chapter 4.5
To rationalize a Single-Term Denominator – Multiply both numerator and denominator by the
radical in the denominator.
1.
2.
To rationalize a denominator with Two Terms – Multiply both numerator and denominator by a
conjugate.
3.
4.
To rationalize an Nth Root Denominator – Multiply by the base raised to the power of the index
minus the exponent.
1
3. 3
9
4.
3
5
3 2
NOTES: ADD AND SUBTRACT RADICALS
Step 1.
Step 2.
Step 3.
Simplify the radicals by adding or subtracting.
19.
21.
32 + 3 2
20.
63 32 ! 53 4
14.
DAY 4
NOTES: SIMPLIFYING WITH VARIABLES
Step 1.
Step 2.
Step 3.
Simplify the radicals.
1.
x 20 2.
x 11 3.
100x 6 4.
32x 13 5.
64x 4 y 100 6.
7.
9.
3 8x 9 y 10
3
16x 11 12 10
8. 3 16x y
10.
4
64x 4 y 100 DAY 5
DAY 1: SIMPLIFYING RADICALS
Simplifying Radicals
1. Approximate the radical as a decimal.
Example(s)
8 5
a. Type into the calculator.
b. Calculate 5 . Then multiply by 8.
2. Find the square root(s).
144
a. Find the number that when you multiply it
by itself, equals the radicand.
Simplifying Square Roots
1. Use the chart to find the largest
perfect square that divides evenly
into the radicand (number under the
radical)
2. Rewrite the radicand as a product
(with one of the factors as the
number you just found)
3. Break up the radical into two (one
with the perfect square and one with
the other factor).
4. Simplify the perfect square.
48
72
DAY 2: SIMPLIFYING NTH ROOTS
Simplifying Nth Roots
3
1. Use the chart to find the largest
perfect power that divides evenly
into the radicand (number under the
radical)
3
250
2. Break up the radical into two (one
with the perfect power)
3. Simplify the perfect root.
Simplifying a radical in the form: a b
1. Simplify the radical.
2. Multiply any like factors together.
3 12
64
DAY 3: OPERATIONS WITH RADICALS
Operations with Radicals
5. Multiplying Radicals
a.
b.
c.
d.
e.
Simplify first, if necessary.
Multiply the radicands.
Place the product under the same radical.
Multiply the “coefficients” if applicable.
Simplify, if necessary.
6. Squares of radicals. Recall,
( 5) = 5
2
a. If there is more than one factor, square each
factor.
Example(s)
a.
7⋅ 3 =
b.
10 2 i 3 8 =
a.
( 10 ) =
b.
(3 5 ) =
a.
7 15 + 15
b.
7 20 - 3 5
c.
2 7 +3 5
a.
16
2
2
2
7. Add/Subtract Radicals.
a. Radicals are considered “Like Radicals” if they
have the exact same radicand (when in
simplified form!).
b. Unlike radicals cannot be combined.
c. Simplify each radical first, if necessary.
d. Add or subtract the “coefficients”.
8. Rationalize the denominator.
Goal: get rid of the radical in the denominator!
a. Simplify any radicals or fractions first.
b. Multiply the numerator and denominator by
the radical in the denominator.
c. Simplify. Recall that
5⋅ 5= 5
b.
21
5 5
DAY 4: RADICAL OPERATIONS
Type
Example
Multiplying Radicals (same root)
1
1. Multiply any coefficients.
2. Multiply the radicands.
3. Keep the same root.
3 12
2 3 10 ⋅ 5 3 4
Dividing Radicals (same root)
2
3
1. Divide the radicands.
2. Keep the same root.
nth Root of a to the nth power
The root and the power cancel out!
Add and Subtract Radicals
6
1. Radicals can only be combined
with addition/subtraction if they
have the same radicand.
2. Simplify each radical separately.
3. Combine the coefficients.
4. Keep the same radical.
7
x7
32 + 3 2
DAY 4: RATIONALIZE THE DENOMINATOR
Single Term Denominator
3
10
5
3 2
Denominator with Two Terms
Nth Root Denominator
7
5
1+ 2
7+3 3
4
3
7
8
3
5
DAY 5: RADICALS WITH VARIABLES
Radicals and Variables
Simplifying Radicals with Variables
1. Find the perfect power that divides
evenly into the coefficient.
Example(s)
a.
a2 =
b.
x6 =
c.
y 12 =
d.
100d 22 =
a.
f 11 =
b.
24 g 51 =
c.
72a 20b 25c 26 =
2. Divide each exponent by the index
(root).
3. Break the radical into two (one that is a
perfect root).
4. Simplify the perfect root.
15. Radicals of fractions.
a. Simplify inside the radical first, if necessary.
Make sure that all exponents are positive.
b. Take the square root of the numerator and the
square root of the denominator.
c. A negative on the outside of the radical
represents -1 multiplied by the radical.
a.
49x 6 y 21
a 15b 100
98 x 6 y 21
b. −
169x −15 y 30