Download Math 1005 Exam 1 Review 1. Perform the indicated operations (a

Math 1005
Exam 1 Review
1. Perform the indicated operations
5 7
5·5+7·2
25 + 14
39
(a) −( + ) = −(
) = −(
)=−
2 5
5·2
10
10
(b)
11 1 11 3 11 · 3 33
÷ =
· =
=
5
3
5 1
5·1
5
3
1 8 1 8 · 2 − 1 · 3 16 − 3 13
(c) ( )−1 − = − =
=
=
8
2 3 2
3·2
6
6
(d)
8 4 8 · 4 32
· =
=
9 5 9 · 5 45
2. State which of the following properties justifies each statement.
Commutative
Identity
Distributive
Associative
Inverse
(a) 4 + (2 + x) = (4 + 2) + x
Associative
(b) 3(x + y) = 3x + 3y
Distributive
(c) x + 0 = x
Identity
(d) xx−1 = 1
Inverse
3. Which of the following
are rational?
q
{
√
2 3
2 , 11 , −3, π,
3 2
11 , 5 ,
−3, and
q
625 2
9 , 5}
√
625
625
√
9 =
9
=
25
3
4. Give an example of a rational number which is not an integer.
1
2
Any fraction would be an example of a rational number which is not an integer.
5. Is the product of √
any two
numbers
always irrational? If no, give an example.
√
√ irrational
√
No,
for
example
2
·
8
=
2
·
8
=
16
=
4
√
√
2 and 8 are irrational, but 4 is rational.
6. Simplify each of the following and write all answers with positive exponents.
(a) x−2 y =
y
x2
(b) (3x4 y−3 )−2 = 3−2 x−8 y6 =
(c) (
y6
y6
=
32 x8 9x8
2
2 −2
x2 y3 −2 2 −2
12
x−4 y−6
2 z−2 ) = ( 2
2 z−2 ) = 4 · 3x z
)
(3x
z
)
=
(
)(3x
)(3x
= 2 6 4
−1
−2
2
4
6
2
4
6
2
2z
2 z
x y z
x y z
x y z
(d) 6x2 y−2 5yx7 = 30x9 y−1 =
30x9
y
7. Evaluate each expression.
√
(a) 3 −27 = −3
Since (−3)3 = −27
4
(b) 64− 3 =
(c)
1
1
1
=√ 4= 4=
3
4
256
64
64
1
4
3
√
3
644 = 44 = 256
8. Write each of the following in simplified radical form.
√
√
√
√
√ √ √
√ √ √
√ √
2x 5
2x 5
20x
2x 5 20x
200x2
100 · 2 · x2 10x 2
2
= √
·√
=
=
=
=
=
(a) √
20x
20x
20x
20x
2
20x
20x
20x
p
p
√
√
√
√
√
√ √
5
5
5
5
(b) 5 96x7 y15 = 5 96 · x7 · 5 y15 = 5 32 · 3 · x x2 · y3 = 2 5 3 · x x2 · y3 = 2xy3 3x2
√
√
√
√
√
√
√
√
√
(c) 3 5 + 3 40 − 3 135 = 3 5 + 3 8 · 5 − 3 27 · 5 = 3 5 + 2 3 5 − 3 3 5 = 0
√
√
√ √
√ √
8y y + 2y x
4 y+ x
2y(4 y + x)
2y
2y
√
√
√
√
=
(d) √
= √
· √
= √
4 y − x 4 y − x 4 y + x (4 y)2 − ( x)2
16y − x
p
p
p
p
√
√
√
(e) 3 250y3 − 3 686y3 = 3 125 · 2 · y3 − 3 343 · 2 · y3 = 5y 3 2 − 7y 3 2 = −2y 3 2
r
√
√
√
5
5
5
8
96x8
32 · 3 · x8 2x 3x3
5 96x
(f)
= √
= √
= √
= 2x
5
5
5
3x3
3x3
3x3
3x3
9. Write each of the following in scientific notation.
(a) 4,390
4.39 × 103
(b) 0.0000325
3.25 × 10−5
10. Refer to polynomials (a) x2 + 4x + x + 4, (b) 5x − 1, and (c) x2 + 2x − 5
(a) What is the degree of (a)? 2 (b)? 1 (c)? 2
(b) Add (a) and (c)
x2 + 4x + x + 4 + (x2 + 2x − 5) = x2 + 4x + x + 4 + x2 + 2x − 5 = 2x2 + 7x − 1
(c) Multiply (b) and (c)
(5x − 1)(x2 + 2x − 5) = 5x3 + 10x2 − 25x − x2 − 2x + 5 = 5x3 + 9x2 − 27x + 5
(d) Subtract (b) from (a)
x2 + 4x + x + 4 − (5x − 1) = x2 + 4x + x + 4 − 5x + 1 = x2 + 5
11. Factor each polynomial completely.
(a) x2 + 5x + 4 = (x + 4)(x + 1)
(b) 3x2 − 14x + 8 = (3x − a)(x − b) = (3x − 2)(x − 4)
(c) 4x2 − 20x + 25 = (2x)2 − 2 · 2x · 5 + 52 = (2x − 5)(2x − 5) = (2x − 5)2
(d) 2x4 − 24x3 + 40x2 = 2x2 (x2 − 12x + 20) = 2x2 (x − 10)(x − 2)
(e) x3 + y3 = (x + y)(x2 − xy + y2 )
(f) 4x2 − 25 = (2x)2 − 52 = (2x − 5)(2x + 5)
(g) x2 y + 2xy2 + x2 y2 = xy(x + 2y + xy)
(h) 2xz+xw−6yz−3yw = (2xz+xw)−(6yz+3yw) = x(2z+w)−3y(2z+w) = (x−3y)(2z+w)
(i) x3 − 3x2 − 9x + 27 = (x3 − 3x2 ) − (9x − 27) = x2 (x − 3) − 9(x − 3) = (x2 − 9)(x − 3)
12. Perform the indicated operation and simplify.
(a)
2x + 8 x2 + 5x + 4 2x + 8
x2 − 9
2(x + 4) (x + 3)(x − 3) 2(x + 3)
÷
=
· 2
=
·
=
2
x−3
x −9
x − 3 x + 5x + 4
x − 3 (x + 4)(x + 1)
x+1
1 2
1 x 2 3
x+6
+
· + ·
x+6
x
x+6
(b) 3 x = 3 x x 3 = 3x =
·
=
1
1 4 x
1 + 4x
3x 1 + 4x 3(1 + 4x)
+4
+ ·
x
x 1 x
x
(c)
(d)
3
4y
3
4y
3 (y − 2)
4y
3(y − 2) − 4y
− 2
=
−
=
·
−
=
=
y + 2 y − 4 y + 2 (y + 2)(y − 2) y + 2 (y − 2) (y + 2)(y − 2) (y + 2)(y − 2)
3y − 6 − 4y
−y − 6
=
(y + 2)(y − 2) (y + 2)(y − 2)
x2 + 7x + 12 x − 4 (x + 3)(x + 4) x − 4 x + 3
·
=
·
=
x2 − 16
2
(x + 4)(x − 4)
2
2