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Math 99 – Practice for 16 Test 3 on Radicals
Test 3 - Practice
1.
Evaluate each radical expression, if possible (If it is not a real number, so state.):
(a)
(e)
2.
1
16
(f)
36
81
4
16 = 2
(c) –  4
(d)
(g) –  25
(h)
3
 64
5
1
(8)
2
11
4
3
(b) 25
1
3
2
(c)
3
3
4
Express each of the following in their simplest exponential form.
(a)
4.
(b)
Evaluate each exponential expression, if possible (If it is not a real number, so state.):
(a)
3.
4
49
3
27x 15
(b)
x7
(c)
x 
5
2
4
5
Simplify the following expressions giving your answer in radical form.
(a)
x6
(b)
3
y 17
(c)
x
5
6
(d)
4
y13
(e)
5
3
 16 z 
1
6  2
5. Simplify (write the answer in radical form without a negative exponent):
6.
 4y 
Simplify (Write exact expressions using radicals, not decimals from a calculator).
Assume the variables represent positive numbers.
(a)
(e)
80
64
81
(b)
(f)
20
25
4
(c)
3
49
(d)
(g)
6
5
(h)
3
32
12
5
Page | 1
Math 99 – Practice for 16 Test 3 on Radicals
7.
Simplify (Write exact expressions using radicals, not decimals from a calculator).
27 x 3 y10 z 5
(b)
2 x 4 y 5 z 6  8 xy 6 z
(c)
(d)
8x 3 y 3 z 5  2 x 2 y 3 z
(e)
64 xy
8x 7 y 5
(f)
(g)
64 xy
8x 7 y 5
(h)
(a)
8.
3
8 yz
3
40 x 6 y 5 z 19
72 x 5 y 4 z 3
(i)
36 x 3 y 5 w 4
(c)
36
49
Simplify by rationalizing the denominator.
(a)
(d)
(g)
12
3
(b)
8x 3 y 4 z 5
3x 2 y 3
8y2 z5
2y
(e)
10 x 2 y
40 xy 5
(f)
(h)
4
4
2


81
81 9
(i)
10 x 3 y 5
2 xy
(b)
8 +
(c)
5  2 5  2 
4 xy
9.
2 y3z
3
8
2
Simplify the following
(a)
12 +
27
20  3 2  2 5
10. Simplify (Write exact expressions using radicals, not decimals from a calculator).
Assume the variables represent positive numbers.
(a)
3
49
(e)
36
49
(i)
3x 2 y 3
4 xy
64
81
(b)
(f)
(j)
3
(c)
4
8x y z
12
3
5
(g)
(k)
6
5
10 x 2 y
40 xy 5
8y2 z5
2y
(d)
(h)
(l)
12
5
8
2
5
10
Page | 2
Math 99 – Practice for 16 Test 3 on Radicals
11. Solve the following Radical Equations.
(a)
x =
(b)
4x  3 =
(c) 2 x
(d)
9
=
9
8
3x  6 = 3
12. Solve the following Radical Equation.
2x  3
=
x
13. Solve the following Radical Equations
2 x  17
x7
=
14. Solve the following Radical Equations.
(a)
4x  5 =
5
(b) 2 3x  2  7
=
1
15. Solve the following Radical Equation
2 x  17
=
x + 1.
16. Find the length of the missing sides in each of the following Right-angled triangles.
c
15
5
12
9
Page | 3
Math 99 – Practice for 16 Test 3 on Radicals
17. Find the length of the missing sides in each of the following Right-angled triangles.
60o
45o
c
c
6
30o
x
45o
x
9
18. Express the following as complex numbers a + bi
(b) 2   25
(a) √
(c)
2 +
8
(e)
4 +
 16
(d) 4 +
 36 +
(f)
9
 25
19. Write each of the following as a complex number in the form a + bi
(a)
 81   36
(b)
(2i – 1)(3i – 2)
20. Write each of the following as a complex number in the form a + bi.
(a) (5 + 4i) + (3 + 2i)
(b) (3 + i) – (7 + 2i)
(c) 2(3 + 2i) + 5(4 – 3i)
(d) 4(3 + 4i) – 2(6 – 6i)
21. Write each of the following as a complex number in the form a + bi.
(a) 2i(3 + 4i)
(b) 5i(3 + 4i)
(c) (2 + i)(4 – 3i)
(d) (5 + 4i)(5 – 4i)
(e) (2 + 3i)2
(f) √
(√
√
)
Page | 4
Math 99 – Practice for 16 Test 3 on Radicals
Test 3 – Practice Solutions
1.
Evaluate each radical expression, if possible (If it is not a real number, so state.):
(a)
49
=
(b)
36
81
=
(f)
1
16
4
(e)
4
=
 64
3
2.
6
9
=
2
3
=
Not real number
–4
=
4
=
4
1
1
1
or –
2
2
=
16
16 = 2
(g) –  25 =
(h)
36
81
(c) –  4
(d)
7 or – 7
1
5
=
Not a real number
–1
Evaluate each exponential expression, if possible (If it is not a real number, so state.):
(a)
(8)
(b) 25
1
2
2
3
=
(8) 2 =
3
1
=
25
1
2
3
64
1
=
25
=
4
=
1
5
=
32 =
11
4
3
(c)
3.
3
3
=
4
11  3
4 4
3
=
3
8
4
9
Express each of the following in their simplest exponential form.
15
(a)
3
27x 15
7
(b)
x
(c)
x 
5
2
4
5
=
3
27  3 x 15
=
3x3
=
3x5
7
2
=
x
=
x4 5 =
5 2
.
10
x 20 =
1
x2
Page | 5
Math 99 – Practice for 16 Test 3 on Radicals
4.
Simplify the following expressions giving your answer in radical form.
(a)
(b)
(c)
(d)
(e)
3
x6
=
x3
y 17
=
y5 3 y2
=
 52 
x 
 
 
x
6
5
4
y13
 4y 
5
6
x15
=
=
y3 4 y
=
43 y 5
 
3
3
64 y15
=
64  y15 =
=
8 y7 y
5. Simplify (write the answer in radical form without a negative exponent):
 16 z 
1
6  2
6.
=
1
=
16 z 
1
6 2
1
16 z 6
=
1
=
16 z 6
 16 z 
1
6  2
1
4z 3
Simplify (Write exact expressions using radicals, not decimals from a calculator).
Assume the variables represent positive numbers.
(a)
80
=
16  5 =
16  5
=
(b)
20
3
49
=
45
=
2 5
=
3
=
4 5 =
3
7
32
=
8 3 4 =
23 4
(c)
(d)
(e)
(f)
(f)
(g)
3
49
64
81
=
25
4
=
6
5
12
5
3
84
=
64
=
8
9
=
5
2
81
25
4
=
=
6 5
=
5 5
12
5
3
=
4 5
30
5
12  5
5 5
=
60
5
=
4 15
=
5
2 15
4  15
=
5
5
Page | 6
Math 99 – Practice for 16 Test 3 on Radicals
7.
Simplify (Write exact expressions using radicals, not decimals from a calculator).
(b)
3
3
3
40 x 6 y 5 z 19
(g)
(h)
(i)
64 xy
8x 7 y 5
2 y3z
8 yz 3
=
=
=
=
16  x 5  y11  z 7
=
4x2 x y5 y z3 z
=
4x2y5z3 xyz
=
3
8  5  3 y 5  3 z 19
=
23 5  x 2  y  z 6  3 z
=
2 x 2 yz 3 3 5z
64 xy
8x 7 y 5
72 x 5 y 4 z 3
16 x 5 y11z 7
40  3 x 6  3 y 5  3 z 19
(e)
(f)
=
3
8x 3 y 3 z 5  2 x 2 y 3 z
=
3xy3z 3 yz 2
2 x 4 y 5 z 6  8 xy 6 z
=
(d)
=
3
= 3xy3 y z 3 z 2
27  3 x 3  3 y10  3 z 5
2 x 4 y 5 z 6  8 xy 6 z =
(c)
(c)
27 x 3 y10 z 5 =
16 x 5 y 6 z 6 =
=
8
8
=
x y4
6
x  y
6
4
72  x5  y 4  z 3
8
=
x y4
8
6
2 y3 z
=
8 yz 3
x6  y 4
y2
4z 2
=
16 x 5
y 6 z 6 = 4x2 x y3z3 = 4x2y3z3
=
2 2
x3 y 2
=
6 2  x 2 x  y 2  z z = 6 x 2 y 2 2 xz
=
2 2
x3 y 2
y2
4  z2
=
x
y
2z
36 x3 y 5 w4  36  x3  y 5  w4  6  x x  y 2 y  w2  6 xy 2 w2 xy
Page | 7
Math 99 – Practice for 16 Test 3 on Radicals
8.
Simplify by rationalizing the denominator.
(a)
12
3

3
3
12
=
3
8y2 z5
(b)
2y
(c)
2y
8 y2 z5

2y
2y
=
36
49
36
=
49
8x 3 y 4 z 5
=
8 x3
(e)
10 x 2 y
40 xy 5
=
x
4y2
(g)
8
2
8
=
3x 2 y 3
(h)
(i)

2
=
2
=
2
3x 2 y 3
4 xy
9.
=
(d)
(f)
12 3
3
=

4 xy
4 xy
4 xy
=
4 3
=
8 y2 z5 2 y
2y
=
4 yz 5 2 y
6
7
2 2 x x y2 z2 z
y4 z5 =
=
x
4 y
8 2
2
=
=
2
=
=
2 xy 2 z 2 xz
x
2y
4 2
3x 2 y 3 4 xy
4 xy
=
3xy 2 4 x y
4
=
6 xy 2 xy
4
=
3xy 2 xy
2
4
4
2


81
81 9
10 x 3 y 5 2 xy 5 x 2 y 4 2 xy
2 xy
10 x 3 y 5 10 x 3 y 5
 5 x 2 y 4 2 xy




1
2
xy
2 xy
2 xy
2 xy
Simplify the following
(a)
12 +
(b)
8 +
(c)
27
=
2 3 3 3  5 3
20  3 2  2 5  2 2  2 5  3 2  2 5   2
5  2 5  2   25  5
2 5 2 2  3
Page | 8
Math 99 – Practice for 16 Test 3 on Radicals
10. Simplify (Write exact expressions using radicals, not decimals from a calculator).
Assume the variables represent positive numbers.
3
49
(a)
3
=
49
64
=
81
(b)
6
5
12
5
(c)
(d)
64
=
81
6 5
=
12
36
=
=
49
=
8 x3
(g)
10 x 2 y
40 xy 5
=
x
4y2
(i)
(j)
2
8
=
2
3x 2 y 3
=

2
4 xy
4 xy
12
=
3
12
3

3
3
8y2 z5
(k)
2y
5
10
(l)
=
2
3x 2 y 3

30
5
12  5
4 xy
4 xy
=
=
2y
8 y2 z5

2y
2y
=
5
10

10
10
60
5
=
5 5
5
8x 3 y 4 z 5
8
8
9
=
(f)
(h)
3
7
=
5 5
=
36
49
(e)
=
4 15
5
=
=
2 15
4  15
=
5
5
6
7
y4 z5 =
x
=
=
4 xy
=
=
=
3xy 2 4 x y
=
4
=
6 xy 2 xy
4
=
3xy 2 xy
2
4 3
8 y2 z5 2 y
2y
5 10
10
2 xy 2 z 2 xz
4 2
3x 2 y 3 4 xy
12 3
3
=
x
2y
=
4 y2
8 2
2
=
2 2 x x y2 z2 z
=
=
4 yz 5 2 y
10
2
11. Solve the following Radical Equations.
x =
(a)
x
=
9
81
(b)
4x  3 =
4x – 3
4x
x
=
=
=
9
81
84
21
Page | 9
Math 99 – Practice for 16 Test 3 on Radicals
2 x
x
x
11.(c)
=
8
=
=
4
16
(d)
3x  6 = 3
3x – 6 = 9
3x = 15
x = 5
12. Solve the following Radical Equation.
2x  3
2x + 3
0
0
=
=
=
=
x
x2
x2 – 2x – 3
(x – 3)(x + 1)
x = 3 or x = – 1
but after checking x = – 1 is unusable its a “phantom solution” so only x = 3 works
13. Solve the following Radical Equations.
2 x  17
2x + 17
2x
x
=
=
=
=
x7
x–7
x – 24
– 24
(after check x = – 24 is not a usable solution so there is no solution to this equation)
14. Solve the following Radical Equations.
(a)
4x  5 =
4x + 5 =
4x =
x =
5
(b) 2 3x  2  7
1
25
2 3x  2
=
8
20
5
3x  2
3x + 2
3x
x
=
=
=
=
4
16
14
15. Solve the following Radical Equation
2 x  17
2x + 17
2x + 17
2x + 17
17
0
0
=
=
=
=
=
=
=
=
2 x  17 =
x + 1.
x+1
(x +1)2
(x + 1)(x + 1)
x2 + 2x + 1
x2 + 1
x2 – 16
(x + 4)(x – 4)
So x = – 4 or x = 4 but after checking x = – 4 is unusable so only x = 4 is a solution
Page | 10
Math 99 – Practice for 16 Test 3 on Radicals
16. Find the length of the missing sides in each of the following Right-angled triangles.
c
5 b
15 c
b
a
a
12
c2
c2
c2
c2
c
c
=
=
=
=
=
=
9
a2 + b2
52 + 122
25 + 144
169
√
13
√
c2 =
152 =
225 =
144 =
=
12 =
a2 + b2\
92 + b2
81 + b2
b2
b
b
17. Find the length of the missing sides in each of the following Right-angled triangles.
60o
45o
c
c
6
30o
45o
x
9
Since this is a 300 – 600 Right Angled Triangle
it has the property that the two arms are connected
and so x = √ 6 this makes the above triangle
look like this.
60o
x
Since this is a 450 – 450 Right Angled
Triangle it has the property that the two
arms are equal so x = 9 this makes the
above triangle look like this.
45o
c
c
6
30o
45o
6√
c2 = a2 + b2
c2 = 62 + (6√ 2
c2 = 36 + 108
c2 = 144
c = √
= 12
18. Express the following as complex numbers a + bi
9
9
c2
c2
c2
c2
c
=
=
=
=
=
a2 + b2\
92 + 92
81 + 81
162
=
√
√
= √
Page | 11
Math 99 – Practice for 16 Test 3 on Radicals
(a) √
=
0 + 3i
(b) 2   25 = 2 – 5i
(c)
2 +
(d) 4 +
9
(e)
4 +
(f)
 36 +
8
=
 16
=
2i  2 2i =
4+
9  1 =
=
=
=
=
0  3 2i
4+
9  1
=
4 + 3i
16  1
4   1 + 16   1
2i + 4i
6i
4  1 +
 25 = 6i + 5i =
0 + 11i
19. Write each of the following as a complex number in the form a + bi
(a)
 81   36
(b)
(2i – 1)(3i – 2) =
=
9i +6i = 15i
6i2 – 4i – 3i + 2 =
– 6 – 7i + 2 =
– 4 – 7i
20. Write each of the following as a complex number in the form a + bi.
(a) (5 + 4i) + (3 + 2i)
(b) (3 + i) – (7 + 2i) =
=
=
5 + 3 + 4i + 2i
8 + 6i
3 + i – 7 – 2i
=
3 – 7 + i– 2i
=
–4–i
(c) 2(3 + 2i) + 5(4 – 3i) =
=
=
6 + 6i + 20 – 15i
9 + 20 + 6i – 15i
29 – 9i
(d) 4(3 + 4i) – 2(6 – 6i)
12 + 16i – 12 + 12i
12 – 12 + 16i + 12i
28i
=
=
=
21. Write each of the following as a complex number in the form a + bi.
Page | 12
Math 99 – Practice for 16 Test 3 on Radicals
(a) 2i(3 + 4i)
(b) 5i(3 + 4i)
=
=
=
=
6i + 8i2
6i + 8(–1)
6i – 8
– 8 + 6i
Using the distributive law
Using the property that i2 = – 1
=
=
=
=
15i + 20i2
15i + 20(–1)
15i – 20
– 20 + 15i
Using the distributive law
Using the property that i2 = – 1
Rearranging terms
Rearranging terms
(c) (2 + i)(4 – 3i)
=
=
=
=
8 – 6i + 4i – 3i2
8 – 2i – 3(–1)
8 – 2i + 3
11 – 2i
Using FOIL
Using the property that i2 = – 1
(d) (5 + 4i)(5 – 4i)
=
=
=
=
25 – 20i + 20i – 16i2
8 – 2(–1)
8+2
10
Using FOIL
Using the property that i2 = – 1
=
=
=
=
=
(2 + 3i)(2 + 3i)
4 + 6i + 6i + 9i2
4 + 12i + 9(–1)
4 + 12i – 9
– 5 + 12i
(e)
(2 + 3i)2
Using the distributive law
Using the property that i2 = – 1
Adding and subtracting like terms
(f) First we convert all the numbers to be in terms of i
√
√
√
(√
=√
=√
√
√
√
)
=
=
=
=
√
√
√
√
√ (√ - √ )
√ √
√ √
i
√
√
– 4 – 12i
Using the distributive law
Using the property that i2 = – 1
Page | 13