Math 209 Team Textbook Solutions Week 4
Show all relevant steps.
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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