Numbers and Exponents
1. Simplify each expression.
a ) ( x8 )( x8 )( x8 )
b) ( x 6 )( x)( x 3 )
c) (−2a )(− a 5b)(a 3b 7 )
d ) (7 x 7 y 5 )(−2 x 4 y 8 )
e) ( xyz 5 )( x 5 y 8 z )
f ) (25 )(28 )(23 )
g ) (−7 a 7b 6 )(2a 5b)
1
h) (4 x 6 y )( x 7 y 8 )
2
i ) (−3 xy 5 z 6 )( xyz 7 )
2. Simplify each expression.
a)
a8b 4 c 3
a 5b 3 c
b)
a 9b
ab
c)
a 8b10
a 2b6
d)
6a 2b
− ab
e)
−72a 3b8
−8ab 6
f)
−54a 6b 4
9a 3b
b)
(10)10 (10)100
(10)108
3. Evaluate each expression.
a)
(4)5 (4)8 (4)10
(4)6 (4)15
(−2)8 (−2)9 (−2)12
c)
(−2)3 (−2)6 (−2)11
4.
(7)2 (7)3 (7)10
d)
(7)14
Evaluate. Leave all of your answers as powers using positive exponents.
a ) 2 −6 × 2 2
b ) 5 −1 × 5 −2
c) 32 ÷ 3−2
d)
(−3)−4
(−3)−2
e) 3−3 × 32 × 3−1
g)
(−3)0
(−3) −1
h)
j)
85 × 8−11
8 −3
k)
32 3−4
×
3−1 30
(−2)9 × (−2) −6
(−2) 2
f)
4 2 −2
×
4 −2 2 2
i)
2 5 4 −1
×
2 3 4 −2
l)
2 7 2 −4
×
2 −2 2 3
5.
Simplify each of the following as far as possible. Leave your answer with positive exponents.
(
)(
a ) a 2b 4 a 2b −5
(
3
d) x y
6.
)
)( x y )
2
2
3
b)
x −6
x −6
(
c)
−1
e) x y
−2
)( x
−2
y
−3
)
x 2 y −2
y −1
x − 3 y −2
f ) 2 −6
x y
Use the laws of exponents to simplify. Leave your answers with positive exponents.
2
5
3
4
a) 5 × 5
1
8
b)
 3 5
c)  10 5 


38
−
3
1
8
3
2
3
d) a ×a
5
4
 −2 2
e)  27 3 


1
 2 −1 2
f )  m3n 4 


7. Express each power as an equivalent radical.
1
1
a) 2 3
b) x 2
d) x
−
3
7
c) 7
−
1
1
e) ( 3 x ) 2
1
2
f ) 3x 2
8. Express each radical as a power.
a)
d)
7
1
( x)
5
4
b)
3
−11
c)
e)
3
2b3
f)
3
n
64
27
9. Express each mixed radical as an equivalent entire radical.
a) 3 2
b) − 4 3
c) 5 27
d) 6 8
e) 2 3 3
f ) 2 4 27
10. Express each entire radical as an equivalent mixed radical.
a)
32
b)
d ) − 6 150
e)
3
48
c) − 3 27
128
f ) 3 3 135
11. Arrange the following from least to greatest by changing to entire radicals: 3 6, 5 2, 2 15, 4 3
Polynomials
1. Expand and collect like terms.
a ) (a + 1)(a + 2)
b) (n − 3)(n − 2)
c) ( x + 9)( x + 7)
d ) (a − 2) 2
e) (2 x − 3)( x − 2)
f ) (2a − 3)(3a + 2)
2. Use the distributive property to determine each product.
a ) n(5n 2 − n + 4)
(
(
c ) ( a − 2 ) a 2 + 2a + 4
b) − k (k 2 − 5k + 1)
d ) (3 p + 2) 5 p2 − 6 p + 2
)
(
)(
e) 2 x 2 + 3 x − 2 5 x 2 + x + 6
)
)
3. Multiply and then collect like terms.
a) ( x − 3)( x + 2 ) + ( 2 x − 5 )
b) ( 2a − 5b )( 3a + b ) − ( 6a − b )( 4a + 7b )
c) 3 ( 2a − 3b )( a + 2b ) − 2 ( 3a − b )
4. A side of a cube is
( x + 2)
2
(
)
(
d ) ( 2 x + 3) 3 x 2 − 5 x + 4 + ( x − 4 ) 2 x 2 − 7
)
cm long. Write a polynomial expression for the volume of the cube and
then expand and collect like terms.
5.
Write a simplified expression for the area of the shaded region:
6n
n
8n
11n
6. Factor the following polynomials.
a) 5 y − 10
b) 51x 2 y + 39 xy 2 − 72 xy
c) 35 z 2 − 14 z 6
d) 3 x 2 + 5 x 3 + x
e) 8 x 2 y − 32 xy 2 + 16 x 2 y 2
7. Factor the following polynomials.
a)
2x ( y − 2) − 3( y − 2)
b) a 2 + 2a + ab + 2b
c) 2 p − 2q + pq − p 2
d) a 2 + 6a + 7 a + 42
8. Factor, if possible.
a) x 2 + 14 x + 40
b) g 2 − 4 g − 77
c) x 2 − 10 x − 24
d) k 2 + 21k + 90
e)
p 2 − 17 p − 60
f)
x 2 + 2 x − 15
g) 2 y 2 − 6 y + 4
h) 6m 2 + 18m − 24
9. Factor each of the following, if possible.
a) 2 x 2 + 3 x + 1
b) 3 y 2 − 2 y − 1
c) 12m 2 − 16m − 4
d) 16 x 2 + 56 xy + 49 y 2
e) 8 x 2 − 29 x − 12
f) 6 y 2 − 11 y − 10
g) 12m 2 + 19m − 10
h) 2 x 2 − 5 + 9 x
10. Determine two values of b that allow each expression to be factored.
a) 3 x 2 + bx + 5
b) 4 x 2 + bx − 9
11. The CN Tower in Toronto has an area that can be expressed as 5 x 2 + 13 x − 6 square units. Factor
5 x 2 + 13 x − 6 to find binomials that represent the length and the width of the tower.
12. The area of a children’s playground measures 2a 2 + 8a − 10 square units.
a) Factor the binomials that represent the length and the width of the playground.
b) If a represents 13 m, what are the length and the width of the playground, in metres.
13. Factor each binomial, if possible.
a) x 2 − 81
b) 4 x 2 − 25 y 2
c) 25 x 2 − 121
d) 81 + x 2
14. Factor each trinomial, if possible.
a)
b)
c)
d)
15.
x 2 − 18 x + 81
x 2 + 14 x + 49
5 x 2 − 10 x + 5
x 2 + 16 x + 64
The area of a square can be given by the expression 9 x 2 − 12 x + 4 , where x represents a
positive integer. Write a possible expression for the perimeter of the square.
Solving Right Triangles
1. A sighting is made of a sailboat from a lighthouse. If the lighthouse is 123.5 m above the level of the
water, how far, to the nearest tenth of a metre, is the sailboat from the shore if the angle of depression is
18°?
2. The light on the Prince Shoal lighthouse is 25 m above the water level. From a position beside the
light, the angle of depression of a sailboat is 12°. How far is the sailboat from the lighthouse, to the
nearest metre?
3. Find YZ, to the nearest tenth of a metre:
4. Find EF, to the nearest tenth of a metre:
5.
Find UV, to the nearest tenth of a metre:
6. From the window of one building, Sam finds the angle of elevation of the top of a second building is
41° and the angle of depression of the bottom of the building is 54°. The buildings are 56 m apart.
Find the height of the second building.
Answer Key
Numbers and Exponents
1a.
x 24
g.
2a.
3a.
−14a12b 7
42 or 16
1
24
5a.
a4
b
5
7
8
7a.
3
8a.
72
1
53
b.
2
10a.
11.
18
3
3
4
( −3 )
x
c.
e.
2
1
1
f. 4 or 22 g. −3 h.
2
3
3
6
25
1
7
5
x y
d.
d. a
d.
b. − 48
d. x
c.
d.
4 2 b. 4 3
c. −9 3
4 3, 5 2, 3 6, 2 15
23
12
e.
e.
x3
−
5
e.
1
7
c. 6 3
675
e.
x6 y9 z6
−3x 2 y 6 z13
d. −6a
e. 9a 2b 2
9
c.
( −2 ) or − 512
4
( −11) 3
−14x11 y13
d.
x2
y
c. 10
1
b.
1
c.
1
b.
1
9a.
c. 34 d.
b.
b.
2a 9b8
c.
h.
2x13 y 9
i.
8
6 4
b. a
c. a b
2
b.
10 or 100
a 3bc 2
4a.
6a.
x10
b.
4
5
1
27
f.
3x
f.
288
d. −30 6
3
1
k. −2
83
l. 2 2
y4
x5
f.
m
1
3
1
3 x
1
( 2b3 ) 3 or 23 b
24
3
e. 4 2
e.
7
n8
1
e.
1
3 5
x y
−6a 3b 3
f.
d.
i. 4 2 or 2 4 j.
216
f.
f.
f.
1
27 n
f.
4
432
3
9 5
Polynomials
1a. a 2 + 3a + 2
b. n 2 − 5n + 6
c. x 2 + 16 x + 63 d. a 2 − 4a + 4 e. 2 x 2 − 7 x + 6 f. 6a 2 − 5a − 6
2a. 5n 3 − n 2 + 4n b. − k 3 + 5k 2 − k
c. a 3 − 8 d. 15 p 3 − 8 p 2 − 6 p + 4 e. 10 x 4 + 17 x 3 + 5 x 2 + 16 x − 12
x 2 + x − 11 b. −18a 2 − 51ab + 2b 2 c. −12a 2 + 15ab − 20b 2
d. 8 x 3 − 9 x 2 − 14 x + 40
3a.
3
4.
( x + 2 ) ⇒ x3 + 6 x 2 + 12 x + 8
5.
6a.
58n 2
5 ( y − 2)
7a.
( 2 x − 3)( y − 2 )
b. 3xy (17 x + 13 y − 24 )
b.
( a + b )( a + 2 )
c. 7 z 2 ( 5 − 2 z 4 )
c.
( 2 − p )( p − q )
d. x ( 3 x + 5 x 2 + 1) e. 8 xy ( x − 4 y + 2 xy )
d.
( a + 6 )( a + 7 )
8a.
e.
9a.
f.
( x + 10 )( x + 4 ) b.
( p − 20 )( p + 3) f.
( 2 x + 1)( x + 1) b.
( 3 y + 2 )( 2 y − 5) g.
( g − 11)( g + 7 ) c.
( x + 5)( x − 3) g.
( 3 y + 1)( y − 1) c.
(12m − 5)( m + 2 ) h.
( x − 12 )( x + 2 )
2 ( y − 2 )( y − 1)
d.
h.
4 ( 3m 2 − 4m − 1) d.
( k + 15)( k + 6 )
6 ( m + 4 )( m − 1)
2
( 4 x + 7 ) e. ( 8 x + 3)( x − 4 )
( 2 x − 1)( x + 5)
10a.
b = ±8 or ± 16
11.
( 5 x − 2 )( x + 3)
b.
12 m × 36 m or
( 2a − 2 )( a − 5) or ( a − 1)( 2a + 10 )
( x + 9 )( x − 9 ) b. ( 2 x − 5)( 2 x + 5) c. ( 5 x − 11)( 5 x + 11) d.
2
2
2
2
b. ( x + 7 )
c.
5 ( x − 1)
d.
( x − 9)
( x + 8)
12a.
13a.
14a.
Trigonometry
1.
2.
3.
4.
5.
6.
380.1 m
117.6 m
YZ = 24.3 m
EF = 332.3 m
UV = 38.6 m
125.8 m
b.
b = 0; ± 5; ± 9; ± 16 or ± 35
24 m × 18 m
Not Possible