Due Date:________________________
Total:
/110
Name:___________________________
Math 10C
Unit 1: Exponents & Radicals Assignment
Section 1a) Number Systems and Prime Factorization ( /36)
1.1 Sort a set of numbers into rational and irrational numbers.
1) Identify each of the following numbers as rational or irrational. Use a calculator to convert
the rational numbers to fractions in simplest form. [4 marks]
1
a) 0.8
9
e) −√1 16
b)√9
c) √0.0064
d) −√79
f) 4.102102102…
g) 5.724 734 744 …
81
h) √√256
1.2 Determine an approximate value of a given irrational number.
1.3 Approximate the locations of irrational numbers on a number line.
1.4 Order a set of irrational numbers on a number line.
2) Order the following irrational numbers on the number line. [2 marks]
a) √14
b) π
c) √80
d) 2√15
1.8 Differentiate between the subsets of the real number system.
3) Find one number which satisfies each condition. [2 marks]
a) An integer, but not a whole number.
__________
b) A rational number, but not an integer.
__________
c) A real number, but not a rational number.
__________
d) A whole number, but not a natural number.
__________
Due Date:________________________
Total:
/110
Name:___________________________
1.9 Represent, using a graphic organizer, the relationship among the subsets of the real numbers
(natural, whole, integer, rational, irrational, real).
4) Use a picture/diagram to represent the six different number sets. Provide an example of each.
[3 marks]
1.10 Determine the prime factors of a whole number.
5) Classify the following whole numbers as prime or composite. [3 marks]
a)20
b) 47
c) 53
d) 87
e) 59
6) Complete the following questions.[3 marks]
a) State the factors of 40:
f) 124
_____________________________
b) State the prime factors of 40:
_____________________________
c) Express 40 as a product of prime factors:
_____________________________
7) In each case write the number as a product of prime factors. [2 marks]
a) 189
b) 690
Due Date:________________________
Total:
/110
Name:___________________________
1.11 Explain why the numbers 0 and 1 have no prime factors.
8) Why the numbers 0 and 1 have no prime factors? [1 mark]
1.12 Determine, using a variety of strategies, the greatest common factor or least common multiple
of a set of whole numbers, and explain the process.
9) State the greatest common factor of [2 marks]
a) 14 and 21
b) 12, 30, and 54
10) Use prime factorization to determine the greatest common factor of [2 marks]
a) 180 and 504
b) 1 700 and 1 938
11) State the lowest common multiple of [2 marks]
a) 4 and 10
b) 9 and 12
12) Determine the lowest common multiple of [2 marks]
a) 6, 10, and 42
b) 12, 30, and 105
Due Date:________________________
Total:
/110
Name:___________________________
13) The lowest common multiple of 35, 231, and 275 is [1 mark]
Response
1.16 Solve problems that involve prime factors, greatest common factors, least common multiples,
square roots or cube roots.
14) A new children’s encyclopedia has 950 pages. Each page contains two background colours for
illustrations. Page 8 and every 8th page thereafter has green as one of the background colours. Page
18 and every 18th page thereafter has orange as one of the background colours. Would this be an
application of greatest common factor or lowest common multiple?
How many pages in the book have both green and orange as background colours? [1 mark]
15) Find the side length of each as an exact value in lowest terms. [3 marks]
a) A square with a perimeter of 40 cm.
b) A square with an area of 40 cm².
c) A cube with a volume of 40 cm³.
16) A cube has a volume of 343 cubic units. Determine the surface area of the cube. [1 mark]
Due Date:________________________
/110
Total:
Name:___________________________
17) Is it possible to construct a square using exactly 200 square tiles? A cube using exactly 200
interlocking cubes? Explain. [2 mark]
Section 1b) Radicals ( /24)
1.13 Determine, whether a given whole number is a perfect square, a perfect cube or neither.
1.14 Determine, using a variety of strategies, the square root of a perfect square.
1.15 Determine, using a variety of strategies, the cube root of a perfect cube.
18) Identify a radical that would be found at each location on the number line. [2 marks]
P: _____
Q: _____
R: _____
S: _____
19) Use estimates to explain why √8 + √17 is not equal to √25.
20) Explain why √4 × √9 × √16 is equal to √576. [1 mark]
𝑛
√𝑥
[1 mark]
Due Date:________________________
Total:
/110
Name:___________________________
1.5 Express a radical as a mixed radical in simplest form (limited to numerical radicands).
1.7 Explain, using examples, the meaning of the index of a radical.
21) Express each radical as a mixed radical in simplest form. [6 marks]
a) √98
b) √108
14
3
d) −5√45
e) 2√88
3
3
3
c) √648
f) 2 √128
4
22) Consider the following radicals 2√8, 3 √1 , 5 √4, 0.5√16. [2 marks]
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.
23) Use the Pythagorean formula, c2=a2+b2, in the given triangle to calculate the length XY.
Express the answer as: [3 marks]
a) an entire radical
b) a mixed radical
c) a decimal to the nearest hundredth
Due Date:________________________
Total:
/110
Name:___________________________
3
3
24) The mixed radical 2√128 can be converted to a mixed radical in simplest form 𝑎 √𝑏. The value
of a+b, to the nearest tenth if necessary, is [1 mark]
1.6 Express a mixed radical as an entire radical (limited to numerical radicands).
1.7 Explain, using examples, the meaning of the index of a radical.
25) Convert the following into entire radicals. [6 marks]
a) 5√3
b) 3√10
3
4
d)−2√6
e) 2√11
3
c) −3√5
14
f) 2 √128
26) Simplify the following radicals then order them from least to greatest. [2 marks]
6
3
4
3
7√1, −3√−27, 2.5√16, 3√ √64
Due Date:________________________
Total:
/110
Name:___________________________
Section 1c) Exponents ( /50)
1.17 Apply the exponent laws:
(am )(an ) = am+n
(ab)m = ambm
am ÷ an = am−n , a ≠ 0
(a/b)m = am/bm, b ≠ 0
(am)n = amn
27) Use the exponent laws to simplify the following. [16 marks]
𝑟4
a) n2 • n5
b) (−2)2 • (−2)5
c)
e) (24 )3
f) (3𝑧 2 )2
g) (−5)2
i) (−7𝑦)
m) 𝑤 −6
0
j) −7𝑦
n)
1
2−3
0
d)
𝑟3
𝑥2
3
𝑚5
𝑚2
h) (−3)0
k) ( )
𝑡
l) (
o) (2𝑚)−3
p)
5𝑚 2
3𝑧 3
𝑚3
𝑛−2
)
Due Date:________________________
/110
Total:
Name:___________________________
28) Combine the laws of exponents to simplify the following: [7 marks]
12𝑑𝑒 3
a) (−2𝑥 4 )(12𝑥 9 )
2𝑑 5 ×𝑑 4
d) (
4𝑑 3
3
b) (
6𝑑𝑒
−5𝑘 3 ×𝑘 2
e) (
)
𝑘
2
c) (−3𝑦 3 𝑧 5 )2
)
𝑘 5 ×𝑘 2
) (
5𝑘 2
)
1.18 Explain, using patterns, why, a-n = 1/an, a≠0
29) State whether the following are true or false. [3 marks]
6
a) 6𝑥 −3 = 𝑥 3
d)
𝑥 −3
2
2
= 𝑥3
1
b) 5𝑎−4 = 5𝑎4
1
e) 5𝑦 −1 = 5𝑦
4
c) 𝑏−6 = 4𝑏 6
1
f) (3𝑥)5 = (3𝑥)−5
30) Simplify the following. Ensure your solution has positive exponents. [5 marks]
a) 𝑎−3 𝑎−3
b) (5𝑏 8 𝑏 −12 )(−10𝑏 3 𝑏 −12 )
c)
16𝑎6 𝑏 −3
−4𝑎6 𝑏 3
Due Date:________________________
−12𝑥 −3
d) (−3𝑎5 𝑏 −3 𝑐 0 )−2
/110
Total:
e) (
6𝑦 −8
Name:___________________________
−1
)
1.19b Express powers with rational exponents as radicals and vice versa using the principle of (1.19a) the
𝑛
pattern of: a1/n= √𝑎 , where a>0.
31) Write an equivalent expression using radicals. [4 marks]
1
a) 𝑏 4
2
b) 𝑓 3
−1
−3
c) 𝑑 2
d) 𝑖 2
32) Assuming that x represent a positive integer, which expressions have no meaning. [2 marks]
7
a) (−𝑥)3
3
1
b) (−𝑥)2
c) (−𝑥)9
5
d) (−𝑥)6
33) Write each of the following as a power and evaluate. [3 marks]
3
a)√64
b) 4
1
√625
c)√√2401
Due Date:________________________
/110
Total:
Name:___________________________
34) Write each expression with positive exponents and simplify. [4 marks]
3
a) √27𝑥 7
4
b) ( √𝑥 3 )(√𝑥)
3
c) √√729𝑦 12
1.20 Solve a problem that involves exponent laws or radicals.
35) A cube has a volume of 729 m3. [2 marks]
a) Determine the side length of the cube.
b) Determine the surface area of the cube.
36) A cube has a volume of V cm3. [2 marks]
a) Write a power and radical which represents the edge length of the cube.
b)Write a power and radical for the area of one of the faces of the cube.
3
d) (√𝑥 3 )( √𝑥 2 )
Due Date:________________________
Total:
/110
Name:___________________________
1.21 Identify and correct errors in a simplification of an expression that involves powers.
37) Find the error in the following question & re-solve to correctly answer the question. [2 marks]