Math 209 Team Textbook Solutions Week 4



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Simplify and reduce to lowest terms.

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Find the root.

−sqrt(25)

= -5

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Find the root. All variables represent nonnegative real numbers.

sqrt(m6 )

= sqrt(m^3 * m^3)

= m^3

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Use the product rule for radicals to simplify the expression. All variables represent nonnegative real numbers.

rt3 ( 5b 9)

= rt3(5 * b^3 * b^3 * b^3)

= b^3 * rt3(5)

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Use the product rule for radicals to simplify the expression. All variables represent nonnegative real numbers.

sqrt ( 8w3y3)

= sqrt(4w^2y^2 * 2wy)

= 2wy sqrt(2wy)

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Simplify the radical. See Example 6. All variables represent positive real numbers.

Sqrt( 9/144)

= sqrt(9)/sqrt(144)

= 3/12 = 1/4

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Evaluate

16 ½

= sqrt(16)

= 4

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Evaluate

1000 2/3

= rt3(1000)^2

= 10^2

= 100

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Simplify. Write your answer with positive exponents. Assume all variables represent positive real numbers.

9 −1 9 ½

= 9 ^ -1/2

= 1 / 9^1/2

= 1/9 * sqrt(9)

= 1/9 * 3

= 3/9

= 1/3

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Simplify the differences. All variables represent positive numbers. See Example 1.

Sqrt 5 − 3 sqrt 5

= -2 sqrt 5

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Simplify. All variables represent positive numbers. See Examples 3 and 4

(3 sqrt2 ) (−4 sqrt 10)

= -12 sqrt 20

= -24 sqrt 5

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Simplify.

sqrt(2t5) * sqrt(10 t4)

= sqrt(20t^9)

= sqrt(4t^8 * 5t)

= 2t^4 sqrt(5t)

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Simplify

(3 – 2 sqrt(7)) (3 + 2 sqrt(7))

Difference of squares:

9 – 4*7

= 9 - 28

= -19

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Write the radical expression in simplified radical form. See Example 2.

Sqrt 2

Sqrt 18

= sqrt(2/18)

= sqrt(1/9)

= 1/3

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Divide and simplify. See Examples 4 and 5.

sqrt 14 ÷ sqrt 7

= sqrt(14/7)

= sqrt(2)

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Simplify. See Example 8.

(3 sqrt 3)4

= 3^4 * sqrt(3)^4

= 81 * 9

= 729

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Simplify

sqrt(6) * sqrt(14)

sqrt(7) sqrt(3)

= sqrt(6*14/7*3)

= sqrt(84/21)

= sqrt(4)

= 2

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Find all real solutions. See Examples 2 and 3.

a2 − 40 = 0

a^2 = 40

a = +/- sqrt(40)

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Solve and check for extraneous roots. Must show check. See Example 4.

Sqrt (a − 1) − 5 = 1

Sqrt(a-1) = 6

a-1 = 36

a = 37

check: sqrt(37-1) – 5 = 6-5 = 1

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Find all real or imaginary solutions to each equation. Use the method of your choice.

3v2 + 4v −1= 0

V = (-4 +/- sqrt(4^2-4*3*-1))/6

V = (-4 +/- sqrt(28))/6

V = (-2 +/- sqrt(7))/3

V = (-2+sqrt(7))/3 or (-2-sqrt(7))/3

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Find all real or imaginary solutions to each equation. Use the method of your choice. Check answers to verify if they are solutions.

Sqrt( 7x + 29) = x + 3

Square:

7x+29 = x^2 + 6x + 9

Subtract:

x^2 - x – 20 = 0

factor:

(x-5)(x+4) = 0

X = 5 or -4

Check: sqrt(7*5+29) = 8 = 3+5, yes!

Check: sqrt(7*-4+29) = 1, so NO

Answer: x = 5

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Solve by using the quadratic equation. See Example 1.

x2 − 7x +12 = 0

x = −b ±sqrt (b^2 – 4 ac)

2a

X = (7 +/- sqrt(7^2-4*1*12))/2

X = (7 +/- sqrt(1))/2

X = (7 +/- 1)/2

X = 3 or 4

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m2+ 2m=8

m^2 + 2m – 8 = 0

factor:

(m+4)(m-2) = 0

M = -4 or 2

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Solve by using the quadratic equation. See Example 3.

p2 + 6p +4 = 0

p = (-6 +/- sqrt(6^2-4*1*4))/2

p = (-6 +/- sqrt(20))/2

p = (-3 +/- sqrt(5))

p = -3+sqrt(5), -3-sqrt(5)

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x2+ 6x + 9=0

(x+3)^2 = 0

X = -3

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Find b2 − 4ac and the number of real solutions to the equation. See Example 5.

−x2 + 3x − 4 = 0

3^2 – 4*-1*-4

= 9 – 16

= -7, no real solutions

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Find all real solutions to each equation.

x2 + x + sqrt( x2 + x) − 2 = 0

z^2 = x^2+x:

z^2 + z – 2 = 0

factor:

(z-1)(z+2) = 0

z = 1 or -2

1 = x^2 + x, so x = (-1+sqrt(5))/2 or (-1-sqrt(5))/2

-2 = x^2 + x has no real solutions

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Find all real and imaginary solutions to the equation. For an imaginary solution, you must use the symbol “i" to mean sqrt(-1)

b4 + 13 b2 + 36 = 0

x = b^2:

x^2 + 13b + 36 = 0

factor:

(x+9)(x+4) = 0

x = -9 or -4

b = +/-sqrt(-9) or +/-sqrt(-4)

b = 3i, -3i, 2i, -2i

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