Objective
Green
Yellow
Red
1.1 Compare and order radical expressions with numerical
radicands.
1.2 Express an entire radical with a numerical radicand as a
mixed radical.
1.3 Express a mixed radical with a numerical radicand as an
entire radical.
1.4 Perform one or more operations to simplify radical
expressions with numerical or variable radicands.
a) Adding and Subtracting
b) Multiplying
c) Dividing
1.5 Rationalize the monomial denominator of a radical
expression.
1.6 Identify values of the variable for which the radical
expression is defined.
1.7 Determine any restrictions on values for the variable in a
radical equation.
1.8 Determine, algebraically, the roots of a radical equation, and
explain the process used to solve the equation.
1.9 Verify, by substitution, that the values determined in solving
a radical equation are roots of the equation.
1.10 Explain why some roots determined in solving a radical
equation are extraneous.
1.11 Solve problems by modelling a situation with a radical
equation and solving the equation.
Math 20-2: Unit 1 Radicals Practice Booklet
Page 1
1a) Radicals Intro and Converting
1. Without using a calculator, determine, where possible, the exact value of the following.
3
a) √49
b) √−4
c) √1000
2. Express each mixed radical as an entire radical.
a) 6√3
b) 2√10
e) 9√6
3
i) −5√3
2
c) −3√2
3
f) 3 √90
g) 2√2
1
d) 2 √24
3
h) 4 √5
13
j) 2 √40
3
3
3
13
3. Consider the radicals 3 √9, 5√2, 2√33, 2 √2080.
a) Explain how to arrange the radicals in order from least to greatest without using a calculator.
b) Arrange the radicals in order from least to greatest without using a calculator.
4. Convert the following radicals to mixed radicals in simplest form.
a) √50
b) √60
c) √54
Math 20-2: Unit 1 Radicals Practice Booklet
d) √108
Page 2
e) 2√75
3
i) √16
f) 2√180
g) −5√162
3
h) 7√63
3
j) √54
3
k) 5√24
l) −3√40
5. The volume of an ice cube is 24000 mm3. The exact length of each edge of the ice cube can be written in
3
simplest mixed radical form as 𝑎 √𝑏 mm, where 𝑎 and 𝑏 are whole numbers. The value of 𝑎 − 𝑏 is
________.
6. Which number is NOT expressed in simplest form?
a) √32
b) √47
c) 7√5
d) 2√13
7. Which number is NOT an entire radical?
a) √47
b) √75
d) √63
c) 2√21
3
8. Which entire radical simplifies to −3√13?
3
3
a) √−351
b) √−117
Math 20-2: Unit 1 Radicals Practice Booklet
3
c) − √117
3
d) √351
Page 3
1b) Adding and Subtracting Radicals
1. After simplifying, are the radicals in each set all like radicals?
a) 6√3, −2√3, 18√3, 4√3
c) 4√5, 12√5, −5√5, √20
b) −7√3, 4√3, √9, 5√3
d) 11√10, −8√10, −√90, √40
2. After simplifying, are the radicals in each set all like radicals?
3
3
3
3
3
3
3
2
a) 2√2, 6√2, 5√2, −4√2
c) 3√6, √24, √48, 12 √6
3
3
3
3
b) 10 √5, 2√5, √5, −5√5
3. Simplify each radical expression.
a) 2√5 − 10√5 + 14√5 − 3√5
3
3
3
3
d) 7 √2, 9√2, √54, − √2
b) 16√2 − 4√2 − 7√2 + 18√2
4. Write each radical expression in simplest form. Then add or subtract.
a) √50 − √8
c) √99 + 3√44
b) √27 + √12
Math 20-2: Unit 1 Radicals Practice Booklet
d) √48 − 2√3
Page 4
5. Simplify each radical expression.
a) √20 + 2√45 − √80
b) √63 + 3√28 − √7
6. Write each radical in simplest form. Then add or subtract.
a) 2√108 − 5√18 − 3√27
b) 8√48 − 2√75 − 6√45
7. A house covers a square with an area of 90𝑚2 and the double garage covers a square with an area of 30𝑚2.
Determine the combined length of the front of the house and garage. Express your answer in simplest
form.
8. An apartment building covers a square with an area of 325𝑚2 and the parking lot behind it covers a square
with an area of 117𝑚2 . Determine the combined length of the apartment building and the parking lot.
Express your exact answer in simplest form.
9. Determine the perimeter of this figure. Express your exact answer in simplest form.
Math 20-2: Unit 1 Radicals Practice Booklet
Page 5
10. Which sets contain like radicals?
a) −√3, √12, 8√3, √27
c) Both A and B
b) 5√8, −√8, 2√32, −√18
d) Neither A nor B
11. Which is the simplest form of 7√2 + √32 − √8 ?
a) 7√26
c) 19√2
b) 7√2 + √8
d) 9√2
12. A triangle has side lengths of 4√3 𝑐𝑚, 5√12 𝑐𝑚, and 2√3 𝑐𝑚. What is the exact perimeter of the triangle,
in simplest form?
13. The sum of any two side lengths of a triangle must be greatest than the length of the third side. Is it
possible to create a triangle with side lengths √175 𝑐𝑚, √63 𝑐𝑚, and √343𝑐𝑚 ? Explain your answer.
Math 20-2: Unit 1 Radicals Practice Booklet
Page 6
1c) Multiplying Radicals
1. Multiply and simplify where possible.
a) (√2)(√11)
b) 4√5 × 2√2
e) (10√2)(√2)
f) (8√3)
2
c) −6√3 × 5√2
d) √7 × √7
g) (−4√7)(−2√3)
h) (9√6)
2. Write the following products in simplest mixed radical form.
a) √8 × √5
b) (√5)(√15)
c) (√6)(√10)
e) (5√18)(2√6)
f) (3√32)(−2√6)
2
d) √8 × √12
g) √15 × 3√27
3. Consider the rectangle shown.
a) Determine the area.
b) Determine the perimeter.
Math 20-2: Unit 1 Radicals Practice Booklet
Page 7
4. Use the distributive property to expand the following.
a) √3(√3 − √7)
b) √5(1 − √5)
c) 2√2(√7 − 3√5)
5. Expand and simplify.
a) √3(√6 − √12)
c) √5(4√5 − √15 + 2√3)
b) √6(√8 − √2)
6. Write and simplify expressions for the area and perimeter of each shape.
a)
b)
7. Expand and simplify.
a) (4√2 + 3)
2
Math 20-2: Unit 1 Radicals Practice Booklet
b) (5√6 − √2)
2
c)(2√3 − √10)(√6 − 7√20)
Page 8
1d) Dividing Radicals and Rationalizing the Denominator
1. Divide the following radicals.
a)
√20
√10
b)
−4√42
b)
√120
√7
c)
8√55
c)
√192
4√6
2√5
d)
−12√30
d)
3√80
8√10
2. Divide and simplify.
a)
√750
√10
√5
2√2
3. Simplify the numerator and denominator, then divide.
a)
4√54
√8
b)
8√126
b)
9√30−3√18
√112
c)
27√24
c)
8√39+8√75
c)
√5
√162
4. Simplify.
a)
√77−√21
√7
3√3
4√3
5. Simplify by rationalizing the denominator.
a)
d)
1
√5
√6
−√7
Math 20-2: Unit 1 Radicals Practice Booklet
b)
3
√3
3
e) √
11
f)
√6
2
5√6
Page 9
6. Express each numerator and denominator in simplest mixed radical form, and then rationalize the
denominator.
a)
d)
5
√50
√147
√98
b)
6
√180
4√75
e) 5√32
c)
√40
√18
20√12
f) 12√20
7. Write each of the following with a rational denominator.
27
a) √ 8
b)
5√14
√70
243
c) √
2
8. Express the following with rational denominators.
a)
√7−√2
√2
b)
√3+2√2
2√3
c)
√5+√2
√6
9. A rectangular garden has length √6 metres and area (9√2 − 6√3) square metres.
a) Write and simplify an expression for the width of the garden.
b) The perimeter of the garden, to the nearest tenth of a metre, is _____.
Math 20-2: Unit 1 Radicals Practice Booklet
Page 10
1e) Radical Expressions with Variables
1. State the values of 𝑥 for which each expression is defined.
a) 5√𝑥 3
b) 7√𝑥 4
c) √𝑥 − 1
d)
10𝑥
√𝑥
2. State any restrictions on the variable. Then simplify the expression.
a) √20𝑥 3
b) √8𝑥 4
3. State any restrictions on the variable. Then simplify the expression.
a) 9√5𝑥 − 7√5𝑥
b) √40𝑥 + 3√10𝑥
4. Determine the following products.
a) √𝑏 × √𝑏
d) (√𝑥)
2
g) √𝑎(√𝑎 − 4√𝑏)
Math 20-2: Unit 1 Radicals Practice Booklet
b) (3√𝑥)(4√𝑥)
e) (3√𝑥)
2
h) 2√𝑥(7√𝑦 − 3√𝑥)
c) 7√𝑎(6√𝑏)
f) (−3√𝑥)
2
i) √𝑥(4 − √𝑥)
Page 11
5. Simplify the expression.
a) (5√𝑥)(4√2𝑥 3 )
b)
d) 2√𝑥(3𝑥 − √𝑥 3 )
e)
12√𝑥 4
c) (√𝑥 + 3)(√𝑥 + 1)
−6√𝑥 3
10
√5
f) 2
√𝑥
√2𝑥
6. For which values of 𝑥 is the expression √5𝑥 − 10 defined?
a) 𝑥 ≠ 2, 𝑥 ∈ 𝑅
b) 𝑥 ≠ −2, 𝑥 ∈ 𝑅
7. For which values of 𝑥 is the expression
a)
𝑥∈𝑅
5
√𝑥+6
a)
9√𝑥
2𝑥
b) 3𝑥 + 1
d) 𝑥 > 2, 𝑥 ∈ 𝑅
c) 𝑥 ≥ 6, 𝑥 ∈ 𝑅
d) 𝑥 > −6, 𝑥 ∈ 𝑅
defined?
b) 𝑥 > 6, 𝑥 ∈ 𝑅
8. Which expression is the rationalized form of
c) 𝑥 ≥ 2, 𝑥 ∈ 𝑅
6+3√𝑥
?
2√𝑥
c)
6√𝑥+3𝑥
2𝑥
d)
6√𝑥+3
2
9. Simplify the expression 8√𝑥 3 (√𝑥 − 3√𝑥 5 )
Math 20-2: Unit 1 Radicals Practice Booklet
Page 12
1f) Solving Radical Equations
1. Describe the first step you would take to solve each equation.
3
a) √6𝑥 = 12
c) √10𝑥 = 20
b) √𝑥 + 3 = 2
d) √4𝑥 − 6 = 13
2. State any restrictions on the variable. Then solve the equation.
a) √𝑥 = 10
d) √2𝑥 − 5 = 3
3
b) √2𝑥 = 6
e) √𝑥 = −4
c) √𝑥 + 7 = 4
f) √𝑥 + 1 = 5
3
3. Consider the equation √𝑥 − 5 = −7
a) State the restrictions on the variable.
b) Solve for 𝑥
c) Verify the solution from part b) . What does this tell you about the solution?
Math 20-2: Unit 1 Radicals Practice Booklet
Page 13
4. Solve the following equations.
𝑥+1
a) √
3
3
4𝑥
b) √
=2
7
=2
5. Solve 1.8 = √0.64𝐴 for A. Leave your answer as an exact value.
3
6. Which of these values is the solution of √𝑥 + 24 = 6?
a) 12
b) -18
c) 192
d) 240
7. A storage tank is in the shape of a cylinder. The radius, r, of the tank is related to the volume, V, according
𝑉
to the formula 𝑟 = √1.2𝜋. Suppose the radius of the tank is 0.9 m. Determine the volume to the nearest
hundredth of a cubic metre.
8. A space station needs to rotate a certain number of times each minute to create the effect of gravity.
Otherwise, the crew are weightless. A formula for determining the number of times a station needs to
rotate to reproduce Earth’s gravity is 𝑁 =
42
𝜋
5
√ , where r is the radius of the station, in metres. Suppose a
𝑟
station rotates 5.6 times per minute. Determine the radius of the space station to the nearest tenth of a
metre.
Math 20-2: Unit 1 Radicals Practice Booklet
Page 14
Unit 1 Radicals Practice Test
1. 6√3 written as an entire radical, is
a) √18
b) √54
c) √108
d) √324
2. Consider the following numbers.
10√6, 4√15, 7√10, 12√5
If the numbers are ranked from largest to smallest, which is the second largest value?
a) 12√5
b) 4√15
c) 7√10
d) 10√6
3. Consider the following statements.
Statement 1: √96 = 4√6
Statement 3: 24 = 4√6
Which of these statements is/are true?
a) 1 only
b) 1 and 2 only
c) 1, 2 and 3
d) Some other combination of 1, 2, and 3
Statement 2: 7√2 = √98
4. When 3√80 + 4√405 is written in the form 𝑘√5, the value of k is
a) 7
b) 48
c) 92
d) 372
5. The area, in cm2 , of a square whose sides are all 4√11 cm is the whole number w. The value of w is
a) 44
b) 176
c) 484
d) 1936
6. The expression √3(√8 − 2√30) can be written in simplest form 𝑎√𝑏 − 𝑐√𝑑 where a, b, c, d are all positive
integers. The value of 𝑎 + 𝑏 + 𝑐 + 𝑑 is _______.
Math 20-2: Unit 1 Radicals Practice Booklet
Page 15
7. √𝑥(4 − √𝑥) is equivalent to
a) 4√𝑥 − √𝑥
b) √4𝑥 − 𝑥
c) 4√𝑥 − 𝑥
d) 4√𝑥 − 2
8. The product (5 − 3√2)(5 + 3√2) is equal to
a) −11
b) 7
c) 19
d) 43 + 6√2
9. 2√3(√243 − 2) − √2(5 + 7√2) can be expanded and simplified to the form 𝑝 + 𝑞√2 + 𝑟√3. The
value of 𝑝 + 𝑞 + 𝑟 is _______.
10. A square is inscribed in a circle as shown. If the radius of the circle is 12 cm, then the exact perimeter of the
square is
a) 12√2 cm
b) 24√2 cm
c) 36√2 cm
d) 48√2 cm
2
11. (2√12 + √24) can be expressed in simplest form as 𝑎 + 𝑏√𝑐. The value of 𝑎𝑏𝑐 is __________.
Math 20-2: Unit 1 Radicals Practice Booklet
Page 16
𝐴
12. If 𝐴 = 15√48 and 𝐵 = 6√150 then 𝐵 is equal to
a) √2
b) 2√2
19√2
c)
42
18√2
d)
13.
6
5
, expressed with a rational denominator, is
√6
a)
1
6
√6
b) √6
c) 6
d) 6√6
14. By rationalizing denominators, the expression
The value of k is _____.
20√5
√10
−
16
√8
can be expressed in the form 𝑘√2, where 𝑘 ∈ 𝑊.
15. Consider the expression √𝑥 − 7. The restrictions on the variable is
a) 𝑥 ≥ 0
b) 𝑥 ≥ 7
c) 𝑥 ≠ 7
d) 𝑥 > 7
3
16. The solution to the equation √2𝑥 = −16 is
a) −2
b) −512
c) −2048
d) Non existent
17. The extraneous root in the radical equation −6 = √30 − 2𝑥 is
a) −3
b) 3
c) 7
d) −7
Math 20-2: Unit 1 Radicals Practice Booklet
Page 17
Answer Key
1a) Radicals Intro and Converting
1.a)7
b) not possible
c) 10
2.a) √108
b) √40
c) −√18
d) √6
3
3
3
h) √320
i) √−375
j) √5
3.a) Turn each into an entire radical and compare the radicands.
4. a) 5√2
5. 17
b) 2√15
h) 21√7
6. A
c) 3√6
3
i) 2 √2
7. C
d) 6√3
3
j) 3 √2
8. A
e) √486
3
3
f) √40
3
13
g) √16
3
b) 3 √9, 5 √2, √2080, 2 √33
2
e) 10√3
3
k) 10 √3
f) 12√5
3
l) −6 √5
g) −45√2
1b) Adding and Subtracting Radicals
1.a) Yes
b) No
c) Yes
d) Yes
2. a) No
b) Yes
c) No
3. a) 3√5
b) 23√2
4. a) 3√2
b) 5√3
c) 9√11
d) 2√3
5. a) 4√5
b) 8√7
6. a) 3√3 − 15√2
b) 22√3 − 18√5
7. 3√10 + √30 𝑚
8. 8√13𝑚
9. 16√5cm
10. C
11. D
12. 16√3𝑐𝑚
13. Yes, a triangle can be formed because the two shorter sides have a sum of 8√7 𝑐𝑚 while the longer is 7√7𝑐𝑚 .
d) Yes
1c) Multiplying Radicals
1.a) √22
b) 8√10
c) −30√6
d) 7
e) 20
f) 192
g) 8√21
h) 486
2. a) 2√10
b) 5√3
c) 2√15
d) 4√6
e) 60√3
f) −48√3
g) 27√5
3. a) 48 𝑢𝑛𝑖𝑡𝑠 2 b) 20√2 𝑢𝑛𝑖𝑡𝑠
4. a) 3 − √21
b) √5 − 5
c) 2√14 − 6√10
5. a) 3√2 − 6
b) 2√3
c) 20 − 5√3 + 2√15
6. a) Area = 22𝑢𝑛𝑖𝑡𝑠 2
Perimeter = 20 𝑢𝑛𝑖𝑡𝑠
2
6. b) Area = 5 + 2√6𝑢𝑛𝑖𝑡𝑠
Perimeter = 4√2 + 4√3 𝑢𝑛𝑖𝑡𝑠
7. a) 41 + 24√2
b) 152 − 20√3
c) 76√2 − 30√15
1d) Dividing Radicals and Rationalizing the Denominator
−3√3
1. a) √2
b) −4√6
c) 4√11
d)
3. a) 6√3
b) 6√2
c) 6√3
4. a) √11 − √3
5. a)
6. a)
7. a)
8.a)
√5
5
√2
2
3√6
4
√14−2
2
b) √3
b)
√5
c)
5
b) √5
b)
c)
3+2√6
6
c)
c)
√30
6
2√5
3
9√6
2
√30+2√3
6
d)
d)
2
−√42
7
√6
2
e)
f)
5√𝑥
𝑥
1f) Solving Radical Equations
1. a) Square both sides
2. a) 𝑥 ≥ 0, 𝑥 ∈ 𝑅 𝑥 = 100
5
d) 𝑥 ≥ , 𝑥 ∈ 𝑅 𝑥 = 7
2
3. a) 𝑥 ≥ 5, 𝑥 ∈ 𝑅
5. 5.0625
Practice Test
1. C
2. D
9. 31
10. D 11. 6912
6. C
b) 2√6
b) 3√10 − √6
c) 2√13 + 10
e)
e)
√33
11
√6
2
9.a) 3√3 − 3√2 m
1e) Radical Expressions with Variables
1.a) 𝑥 ≥ 0, 𝑥 ∈ 𝑅
b) 𝑥 ∈ 𝑅
c) 𝑥 ≥ 1, 𝑥 ∈ 𝑅
2. a) 𝑥 ≥ 0, 𝑥 ∈ 𝑅 2𝑥√5𝑥
b) 𝑥 ∈ 𝑅
2𝑥 2 √2
4. a) 𝑏
b) 12𝑥
c) 42√𝑎𝑏
d) 𝑥
h) 14√𝑥𝑦 − 6𝑥
i) 4√𝑥 − 𝑥
5. a) 20𝑥 2 √2
b) −2√𝑥
√10𝑥
2𝑥
2. a) 5√3
7. D
f)
f)
c) √2
d) 3√10
√6
15
√15
3
b) 6.8
d) 𝑥 > 0, 𝑥 ∈ 𝑅
3. a) 𝑥 ≥ 0, 𝑥 ∈ 𝑅 2√5𝑥
b) 𝑥 ≥ 0, 𝑥 ∈ 𝑅
e) 9𝑥
f) 9𝑥
g) 𝑎 − 4√𝑎𝑏
5√10𝑥
c) 𝑥 + 4√𝑥 + 3 d) 6𝑥 √𝑥 − 2𝑥 2
8. C
9. 8𝑥 2 − 24𝑥 4
b) square both sides
c) cube both sides
d) add 6 to both sides
b) 𝑥 ≥ 0, 𝑥 ∈ 𝑅 𝑥 = 18
c) 𝑥 ≥ −7, 𝑥 ∈ 𝑅 𝑥 = 9
e) 𝑥 ∈ 𝑅
𝑥 = −64
f) 𝑥 ≥ −1, 𝑥 ∈ 𝑅 𝑥 = 124
b) 𝑥 = 54
c) 𝑥 = 54 is an extraneous root
4. a) 𝑥 = 11
b) 𝑥 = 14
6. C
7. 𝑣 = 3.05𝑚3
8. 𝑟 = 28.5 𝑚
3. B
12. A
Math 20-2: Unit 1 Radicals Practice Booklet
4. B
13. B
5. B
14. 6
6. 24
15. B
7. C
16. C
8. B
17. A
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