Name:_Solutions_____________________________________________ Date:_03 February 2017____
Quiz Rational & Irrational Numbers [50 points]
Make sure to show all of your work and use complete sentences when necessary. Partial credit will be
given but ONLY if you have work shown.
1. State the following definitions and give two examples for each of the following: [6 points]
a. Natural Numbers
Natural numbers are considered the counting numbers; they are also all positive, whole
numbers. Examples include: 1,2,3,4,5 ….
b. Integers
Integers are all whole numbers. The set of these numbers are: {… , −2, −1, 0, 1, 2, … }
c. Rational Numbers
𝑝
𝑞
Any number that can be represented as a ratio of two integers , 𝑞 ≠ 0. In decimal form,
2
3
the decimal places will either end or repeat. Examples: , −3, 1.673, 0. 8�
d. Irrational Numbers
Any number that cannot be represented as a ratio of two integers. In decimal form, the
decimal will go on forever and not repeat. Examples: −√3, √2, 𝜋
2. Solve for the missing variable. Pick and solve two of the four problems given. Circle the letter for
the problem you are solving for. [2 points]
a. 5𝑥 = 1 → 𝑥 = 𝟎
b. −32 = 𝑥 → 𝑥 = −1 ∙ 32 = −1 ∙ 9 = −𝟗
c. 5−1 = 𝑥 → 𝑥 = 𝟏/𝟓
d. (𝑥)3 = −27 → 𝑥 = 𝟑
3. Decide whether each number is Rational or Irrational. Label with an R or an I in the box below.
Then, explain how you know using at least one complete sentence. Pick an solve up to 10 points
of problems. [10 points]
Number
5/9
+1
+1
+1
+1
Explanation
5/9 is being expressed as a ratio of two integers which is
the definition of a rational number
√6
Irrational
The square root of 6 does not come out to equal an
integer value, thus meaning the decimal will never end
nor repeat.
√49
Rational
√49 = 7. This is an integer value which means it is a
rational number.
0. 9�
Rational
The line above the 9 means that it is a repeating value,
thus part of the definition of a rational number.
1.414213
Rational
The decimal places ends. Although these are the first 6
decimal points of √2, there is nothing that indicates that
the decimal points will continue.
Rational
√5 × √20 = √100 = 10
Rational
√9 + 3 + √49 = 3 + 3 + 7 = 13
+1
√5 × √20
+1
√9 + 3 + √49
+2
+2
Rational OR
Irrational
Rational
√21
Irrational
√3 + √3
Irrational
3 − √7
Irrational
√9
+2
3𝑐 + √7𝑐 2
Where c is a
positive
integer +3
√21
√9
=�
21
9
= √3. This value is irrational.
By definition, the sum of two irrational numbers is
always irrational
3−√7
3𝑐+√7𝑐 2
3−√7
.
√7�
= 𝑐�3+
Here nothing cancels out, thus this
value is irrational.
4. Answer 4 of the following 5 problems concerning statements about rational and irrational
numbers with geometric shapes. [12 points]
a. Calculate the perimeter and area of this rectangle. Then circle the R if the perimeter
and area is rational or I if they are irrational.
2 + √7
5 − √7
Perimeter: _14_______ R / I
Area: _3 + 3√7_______ R / I
b. Find values for the rectangle such that the perimeter is an irrational number and the
area is a rational number.
Example:
𝑆𝑖𝑑𝑒 𝐿𝑒𝑛𝑔𝑡ℎ𝑠: √3 × √12
𝐴𝑟𝑒𝑎 = √3 × √12 = √36 = 6
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 2√3 + 2√12
c. Find values such that the product of two irrational numbers is rational.
Solutions Will Vary. Example:
√2 1
∙
1 √2
=
√2
√2
=1
d. True or False: Between any two rational numbers there is an irrational number line. Give
an example to show.
True. Example:
3 𝜋
4
e. Find values such that the sum of two irrational numbers is rational.
Solutions will vary. Example: 𝜋 + (−𝜋) = 0
5. Simplify the following rational expressions: [8 points]
a.
𝑥2𝑦9𝑧 3
b.
(𝑥𝑦)2 (𝑥 2 𝑦 3 )4
c.
(𝑥𝑦)−3
=
𝑥𝑦 12 𝑧 3
𝑥5
𝑥𝑦 −2
=
𝑥 2−1 𝑧 3−3
𝑦 12−9
=
=
𝒙
𝒚𝟑
𝑥 2 𝑦 2 𝑥 8 𝑦 12
𝑥 −3 𝑦 −3
𝑥𝑦 −2
𝑥5
=
𝑦2
𝑥∙𝑥 3 𝑦 3
=
=
6. Simplify two of the following: [6 points]
a. 93/2 = 33 = 𝟐𝟕
b. 363/2 = 63 = 𝟐𝟏𝟔
c. 272/3 = 32 = 𝟗
𝑥 10 𝑦 14
𝑥5
𝟏
𝒙𝟒 𝒚
= 𝒙𝟓 𝒚𝟏𝟒
7. Eric is painting a large canvas with dimensions √12 feet by √3 feet. He has enough paint to
cover 5 square feet of canvas. Does he have enough paint to paint the entire canvas? Show why
or why not. [6 points]
Area of the painting is √12 ∙ √3 = √36 = 6 𝑓𝑡 2 . Because 6 𝑓𝑡 2 > 5 𝑓𝑡 2 , Eric does not have
enough paint to cover the canvas.
8. Bonus: Given that 5𝑎 = 3, use the laws of exponents to write a number that is equivalent to
each expression. [3 points]
a. 52𝑎 =
52𝑎 = (5𝑎 )2 → 32 = 𝟗
b. 5𝑎+1 =
5𝑎+1 = 5𝑎 ∙ 51 = 5𝑎 ∙ 5 → 3 ∙ 5 = 𝟏𝟓
c. 31/𝑎 =
31/𝑎 = (3)1/𝑎 → (5𝑎 )1/𝑎 = 5𝑎/𝑎 = 𝟓
9. Bonus: Simplify the following expression: [2 points]
4
𝑥 1/2 𝑦 −3
𝑥 2 𝑦 −12
� −9/2 −5 � = −18 −20 = 𝒙𝟐𝟎 𝒚𝟖
𝑥 𝑦
𝑥
𝑦