simplifying radicals index radical radicand steps for simplifying square roots 1. factor the...
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Simplifying Radicals
n aIndexRadical
Radicand
Steps for Simplifying Square Roots 1. Factor the Radicand Completely or until you find a perfect root
2. Take out perfect roots (look for pairs)
Note: With square roots the index isnot written
3. Everything else (no pairs) stays under the radical
Root Properties:
[1] 00 n
[2] If you have an even index, you cannot take roots of negative numbers. Roots will be positive.
answer real-non 4
[3] If you have an odd index, you can take the rootsof both positive and negative numbers. Roots may beboth positive and negative
3 273 3 273
General Notes:
[1] 416 4 is the principal root
[3] 416 ±4 indicates both primary and secondary roots
[2] 416 – 4 is the secondary root(opposite of the principal root)
Radicals CW12428 yx[1] 2410256 yx[2]
101720 yx[3] 154121 yx[4]
1014162 yx[5] 12445 yx[6]
12414 yx[7] 171056 yx[8]
101778 yx[9] 134225 yx[10]
51227 yx[11] 111481 yx[12]
72 62yx12516 yx
xyx 52 58 yyx 7211
29 57 yx 53 62yx
1462yx yyx 142 85
xyx 7858 yyx 6215
yyx 33 26 yyx 579
Radicals
Simplifying Cube Roots (and beyond)
1. Factor the radicand completely
2. Take out perfect roots (triples)
Example 1
a] 3 54 b] 3 24
Example 4 Applications Using Roots
[A] The time T in seconds that it takes a pendulum to make acomplete swing back and forth is given by the formula below, where L is the length of the pendulum in feet andg is the acceleration due to gravity. Find T for a 1.5 footpendulum. Round to the nearest 100th and g = 32 ft/sec2.
g
LT 2
seconds 36.132
5.12
T
Example 5 Applications Using Roots
[B] The distance D in miles from an observer to the horizonover flat land or water can be estimated by the formulabelow, where h is the height in feet of observation. Howfar is the horizon for a person whose eyes are at 6 feet?Round to the nearest 100th.
hD 23.1
miles 01.3
623.1
D
Simplifying Radicals
Example 1 Multiplying Radical Expressions
[A]
3253 [C] [D] 3325
327 [B] 235
5215
6325315
1. Multiply radicand by radicand
2. If it’s not underneath the radical then do not multiply, write together (ex: )32
1525336
Example 2 Foil
a] )4)(2( xx b] )7)(5( mm
mmm 7535
mm 1235
c] )43)(26( xx d] )45)(23( xx
8101215 xxx
8215 xx
Example 3 Simplify Sums / Differences•Find common radicand•Combine like terms
a] 3273122 b] 2218385
2229210
23
Conjugate: Value that is multiplied to a radical expressionThat clears the radical.
Rationalizing: Multiplying the denominator of a fraction by its conjugate.
[1] [2]
[3] [4]
[5] [6]
[7] [8]
[9] [10]
[11] [12]
Simplifying Radicals
237
5
1
453
3213 1515
7
3
3y
x5
3
x
37
5
x3
14
10
y
3114
2
yx5
15
xy
Simplifying Radicals
Binomial Conjugate: Binomial quantity that turns the expressioninto a difference of squares.
y2 y2 y4 x5 x5 x25
Rational Exponents Property: n
mn m xx
Example 1: Rational to Radical Form
A] 3
2
x B] 5
4
x C] 2
1
x
3 2x5 4x x
Radicals
Radicals CW
3 5x
Write in rational form.
1. 2. 5 7x 3. 5x 4. 7 3x
3
1
x5. 6. 5
2
x 7. 2
5
x 8. 7
3
x
Write in radical form.
Radicals
Radical Equation Equation with a variable under the radical sign
Extraneous Solutions Extra solutions that do not satisfy equation
Radical Equation Steps[1] Isolate the radical term (if two, the more complex)
[2] Square, Cube, Fourth, etc. Both Sides
[3] Solve and check for extraneous solutions
Radicals CW
171122 x
4106 x
1711153 x
Solve Algebraically.
9. 10.4125 x
11. 12.
165 x2565 x251x
146 x1966 x190x
622 x3622 x382 x
19x
6153 x215 x415 x55 x1x
Radicals CW
95124 x4124 x
162147 x
Solve Algebraically.
13. 14.
112 x112 x02 x0x
14147 x
214 x414 x34 x
4
3x