Exam #3 Review
Math 10 Fall 2012
Name:
For attendance, turn in solutions to any two of the problems marked with a F.
Section 4.5 / Appendix B
Part 1
Determine if the ordered triple (5, −3, −2) is a solution of the system:
x +
y +
z =
0
x + 2y − 3z =
5
3x + 4y + 2z = −1
Part 2
Solve the following system. If there is no solution or if there are infinitely many solutions
and the system’s equations are dependent, so state:
2x +
y − 2z = −1
3x − 3y −
z =
5
x − 2y + 3z =
6
1
Part 3
Solve each system using matrices. If there is no solution or if there are infinitely many
solutions and a system’s equations are dependent, so state:
(b)
(a)
x +
y +
3x −
z = 3
−6x + 2y − 4z = 1
−y + 2z = 1 F
−x
+
y + 2z = 4
5x − 3y + 8z = 0
z = 0
2
Chapter 10, Sections 1-7
Part 1
Simplify each expression:
√
(a) 72
(b)
p
(−10)2
√
(d) − 49x6
√
(c)
(e)
qp
√
3
169 +
√
x2 + 14x + 49
9+
p√
3
1000 +
√
3
216F
Part 2
Use rational exponents to simplify each expression. If rational exponents appear after
simplifying, write the answer in radical notation. Assume that all variables represent
positive numbers:
p
√
√ √
3
5
y2
(b) 3 · 3 3
(a) 27a12
p
(c) 10
y3
(d)
p
√
5
x
−6 3
(e) (8x y )
3
1
3
5
6
x y
− 13
6
Part 3
Simplify by factoring:
√
(a) 45
(c)
p
3
(b)
−32x2 y 3
(d)
√
40x
p
5
64x7 y 16
Part 4
Multiply and simplify. Assume that all variables in a radicand represent positive real
numbers and no radicands involve negative quantities raised to even powers:
p
p
√
√
(b) 5 8x4 y 3 z 3 · 5 8xy 9 z 8 F
(a) 2x7 · 12x4
Part 5
Perform the following operations:
p
p
(a) 4 3 x4 y 2 + 5x 3 xy 2
√
p
9x2 64y
2
3
(b) 5 8x y − p
3x 2y −2
4
Part 6
Rationalize each denominator. Simplify, if necessary:
25
(a) p
5x2 y
(b) √
17
10 − 2
Part 7
Add or subtract as indicated. Begin by rationalizing denominators for all terms in which
denominators contain radicals:
r
r
√
√
5
√
5
3
√ − 2 32 + 28F
(b) √
+
(a) 15 −
2+ 7
3
5
Part 8
√
Let f (x) = x2 + 4x − 2. Find f (−2 + 6).
5
Part 9
Solve each radical equation:
√
(a) x = 3x + 7 − 3
1
(b) (2x + 3) 4 + 7 = 10
Part 10
1
1
If f (x) = (9x + 2) 4 and g(x) = (5x + 18) 4 , find all values of x for which f (x) = g(x).
Part 11
Write each expression in the form a + bi:
√
(a) − −300
(b) (8 − 5i) − (6 + 2i)
√
√
(d) (5 − i 3)(5 + i 3)
(c) (5 − 3i)2
(e)
√
−3 ·
√
−36
(f)
6
5i
2 − 3i
Section 11.1
Part 1
Solve each quadratic equation by completing the square:
(a) x2 + 6x = 7
(b) x2 + 8x − 5 = 0
(c) 3x2 − 6x + 2 = 0
(d) 9x2 − 6x + 5 = 0F
7
Miscellaneous
Part 1
In Chapter 7, we learned the Fundamental Principle of Rational Expressions,
PR
P
= .
QR
Q
However, a common mistake that students make when simplifying a fraction is dividing
out common terms from the numerator and denominator rather than common factors.
To show why this is incorrect, find real numbers a, b, and c such that c 6= 0 and
a + bc
6= a + b.
c
Part 2
For what real numbers a, b is (a + bi)(a − bi) real and nonnegative?
Part 3
For what real numbers a, b is (a + bi)(a − bi) = 0?
8