NOTES: SIMPLIFYING SQUARE ROOTS DAY 1 - lcps.org · To rationalize an Nth Root Denominator – Multiply by the base raised to the power of the index minus the exponent. 3. ... square

NOTES: SIMPLIFYING SQUARE ROOTS DAY 1 Simplify the Square Roots. Step 1. Step 2. Step 3.

1. 50 2. 200

3. 48 4. 128

5. 2 54 5. 2 48

NOTES: SIMPLIFYING CUBE ROOTS DAY 1 Step 1. Step 2. Step 3. Simplify the Cube Roots. 3.

6.

3. 5 643

16

3

NOTES: SIMPLIFYING NTH ROOTS DAY 2 Step 1. Step 2. Step 3. Simplify the Cube Roots. 1. 3 2564

2.

3. 3 324 4. 10 325

5. (x + 4)33 6.

x77

x77

NOTES: MULTIPLY AND DIVIDE RADICALS DAY 3 Step 1. Step 2. Step 3. Radical Operations: Multiply, Divide

11. 8 i 2 12. 3 43 i 5 23

13.

753

16. 3 i 5

17. 18.

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.

3. 4.

1

93

3

523

NOTES: ADD AND SUBTRACT RADICALS DAY 4

Step 1. Step 2. Step 3. Simplify the radicals by adding or subtracting. 19.

20.

21. 14.

32 + 3 2

6 32

3! 5 4

3

NOTES: SIMPLIFYING WITH VARIABLES DAY 5 Step 1. Step 2. Step 3. Simplify the radicals.

1. x 20 2.

3. 100x 6 4.

5. 64x 4 y100 6. 16x 113

7. 8x 9 y103 8. 16x12 y103

9. 10. 64x 4 y1004

x11

32x

13

DAY 1: SIMPLIFYING RADICALS

Simplifying Radicals Example(s)

1. Approximate the radical as a decimal.

a. Type into the calculator. b. Calculate 5 . Then multiply by 8.

8 5

2. Find the square root(s).

a. Find the number that when you multiply it by itself, equals the radicand.

144

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

1. Use the chart to find the largest perfect power that divides evenly into the radicand (number under the radical)

2. Break up the radical into two (one

with the perfect power)

3. Simplify the perfect root.

2503

643

Simplifying a radical in the form:

1. Simplify the radical. 2. Multiply any like factors together.

a b

3 12

DAY 3: OPERATIONS WITH RADICALS

Operations with Radicals Example(s)

5. Multiplying Radicals

a. Simplify first, if necessary. b. Multiply the radicands. c. Place the product under the same radical. d. Multiply the “coefficients” if applicable. e. Simplify, if necessary.

a. 7 3⋅ = b. i10 2 3 8 =

6. Squares of radicals. Recall, ( ) =2

5 5

a. If there is more than one factor, square each factor.

a. ( ) =2

10

b. ( ) =2

3 5

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”.

a. 7 15 + 15 b. 7 20 - 3 5 c. 2 7 + 3 5

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⋅ =

a. 162

b. 215 5

DAY 4: RADICAL OPERATIONS Type Example

3 12

1

Multiplying Radicals (same root)

1. Multiply any coefficients. 2. Multiply the radicands. 3. Keep the same root. 2 103 ⋅ 5 43

2

Dividing Radicals (same root)

1. Divide the radicands. 2. Keep the same root.

3 nth Root of a to the nth power The root and the power cancel out! x77

6

Add and Subtract Radicals

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.

32 + 3 2

DAY 4: RATIONALIZE THE DENOMINATOR

Single Term Denominator

3

5

10

3 2

Denominator with Two Terms

7

1+ 2

5

7+ 3 3

Nth Root Denominator

4

73

8

53

DAY 5: RADICALS WITH VARIABLES Radicals and Variables Example(s)

a. 2a =

b. 6x =

c. 12y =

d. 22100d =

Simplifying Radicals with Variables

1. Find the perfect power that divides evenly into the coefficient.

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.

a. 11f =

b. 5124 g =

c. 20 25 2672a b c =

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. 6 21

15 100

49x ya b

b. 6 21

15 30

98169

x yx y−−