Name
Class
Date
Reteaching Roots and Radical Expressions For any real numbers a and b and any positive integer n, if a raised to the nth power equals b, then a is an nth root of b. Use the radical sign to write a root. The following expressions are equivalent: index
power an = b
·
radicand n
1b = a
radical sign Problem
What are the real-number roots of each radical expression? 3 a. 1 343
Because (7)3 = 343, 7 is a third (cube) root of 343. 3 Therefore, 1343 = 7. (Notice that (-7)3 = -343, so -7 is not a cube root of 343.)
4 1 b. 5 625
1 Because 1 15 24 = 625 and
3 c. 1 -0.064
Because (-0.4)3 = -0.064, -0.4 is a cube root of -0.064 and is, in fact, the only one. 3 So, 1-0.064 = -0.4.
d. 1-25
Because (5)2 = (-5)2 = 25, neither 5 nor -5 are second (square) roots of -25. There are no real-number square roots of -25.
1 1 1 - 15 24 = 625 , both 15 and - 15 are real-number fourth roots of 625 .
Exercises Find the real-number roots of each radical expression. 3
4
1. 1169
2. 1729
3. 10.0016
4. 5- 18
4 5. 5 121
6. 5216
3
4 7. 5 - 25 4
10. 1 -0.0001
4
8. 10.1296 1 11. 5243 5
3
125
3
9. 1 -0.343 3
8
12. 5125
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Name
Class
Date
Reteaching (continued) Roots and Radical Expressions n
You cannot assume that 2an = a. For example, 2(-6)2 = 136 = 6, not -6. This leads to the following property for any real number a: n
2an = a
If n is odd
n
2an = 0 a 0
If n is even Problem
What is the simplified form of each radical expression? 3
a. 21000x 3y 9
3 3 2 1000x3y 9 = 2 103x3(y 3)3 3 =2 (10xy 3)3
= 10xy 3
Write as the cube of a product. Simplify.
8
b. 5 256g h4k 16 4
Write each factor as a cube.
4
4 44(g 2)4 256g 8 = 5 h4(k 4)4 h4k 16
Write each factor as a power of 4.
5
2
4 4g =5 a 4b
hk
=
4g 2 0 h 0 k4
4
Write as the fourth power of a quotient. Simplify.
The absolute value symbols are needed to ensure the root is positive when h is negative. Note that 4g2 and k4 are never negative.
Exercises Simplify each radical expression. Use absolute value symbols when needed. 13. 236x 2 16.
2x20 2y 8
14. 2216y 3
15. 5 1 2 100x
3 3 17. 5 (x + 3) 6 (x - 4)
18. 2x10y 15z 5
3
5
3 27z 3 19. 5 (z + 12)6
20. 22401x 12
3 21. 51331 x3
8 4 22. 5 (y - 4) (z + 9)4
3 6 6 23. 5 a b c3
24. 2 -x3y 6
4
3
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Name
Class
Date
Reteaching Multiplying and Dividing Radical Expressions You can simplify a radical if the radicand has a factor that is a perfect nth power and n is the index of the radical. For example: n
n
1xy nz = y 1xz Problem
What is the simplest form of each product? 3 a. 1 12
# 110 112 # 110 = 112 # 10 = 222 # 3 # 2 # 5 = 223 # 3 # 5 = 223 # 23 # 5 3
3
3
#
n n n Use 1a 1b = 1ab .
3
3
Write as a product of factors.
3
Find perfect third powers.
3
n
#
n
n Use 1ab = 1a 1b .
3
n
Use 2an = a to simplify.
3 = 21 15
# 221xy2 27xy 3 # 221xy 2 = 27xy 3 # 21xy 2 = 27xy 2y # 3 # 7xy 2 = 272x2(y 2)2 # 3y
b. 27xy 3
#
n n n Use 1a 1b = 1ab .
Write as a product of factors. Find perfect second powers. n
Use 1an = a to simplify.
= 7xy 2 23y
Exercises Simplify each product.
# 135x 4. 5 27x 3y # 228y 2 1. 115x
# 220y 5. - 29x 5y 2 # 22x 2y 5 3
2. 250y 2 3
3
3
# 2 -6x2y 6. 13 ( 112 - 121 ) 3
3. 236x 2y 5
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3
Name
Class
Date
Reteaching (continued) Multiplying and Dividing Radical Expressions Rationalizing the denominator means that you are rewriting the expression so that no radicals appear in the denominator and there are no fractions inside the radical. Problem
What is the simplest form of
19y ? 12x
Rationalize the denominator and simplify. Assume that all variables are positive. 19y 9y = 12x 5 2x
Rewrite as a square root of a fraction.
# 2x # 2x
9y
=5 2x
Make the denominator a perfect square.
18xy
=5 2 4x =
Simplify.
118xy 122 x 2
#
=
218xy 2x
=
232
=
3 22xy 2x
Write the denominator as a product of perfect squares. Simplify the denominator.
#2#x#y
Simplify the numerator.
2x
n
Use 1an = a to simplify.
Exercises Rationalize the denominator of each expression. Assume that all variables are positive. 7.
11.
15 1x 3
4 2k 9 3 161k 5
8.
3
26ab2 3 12a4b
5 12. 5 3x 5y
4
9.
13.
29y 4 1x 4
210 4 1z 2
10.
14.
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210xy 3 112y 2 19a2b 7 abc4 3
Name
Class
Date
Reteaching Binomial Radical Expressions Two radical expressions are like radicals if they have the same index and the same radicand. Compare radical expressions to the terms in a polynomial expression. Like terms:
4x3
11x3
Unlike terms:
4y 3
11x3
Like radicals:
3 41 6
Unlike radicals:
3 41 5
3 11 1 6
The power and the variable are the same 4y 2
3 11 1 6
Either the power or the variable are not the same. The index and the radicand are the same
2 41 6 Either the index or the radicand are not the same.
When adding or subtracting radical expressions, simplify each radical so that you can find like radicals. Problem
What is the sum? 163 + 128
# 7 + 14 # 7 = 232 # 7 + 222 # 7
163 + 128 = 19
Factor each radicand. Find perfect squares. n
#
n
= 232 27 + 222 27
n Use 1ab = 1a 1b.
= 517
Add like radicals.
n
Use 2an = a to simplify.
= 317 + 217
The sum is 517.
Exercises Simplify. 1. 1150 - 124 3
3
4. 512 - 154
3
3
2. 1135 + 140
5. - 148 + 1147 - 127
3. 613 - 175
3 3 3 6. 81 3x - 1 24x + 1 192x
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