Classify It!
Which number is an integer?
A. −8
B. 2.6
C. 3/7
D. sqrt11
A. −8. Integers include negative whole numbers, zero, and positive whole numbers.
Evaluate √169.
Answer: 13, the principal (nonnegative) square root.
Evaluate ∛27.
Answer: 3
3³ = 27.
Which number is irrational?
A. √36 B. −2.4 C. √19 D. 5/8
Answer: C. √19.
Irrational because the value cannot be expressed as a fraction. The decimal is a non-terminating, non-repeating decimal.
Between which two consecutive whole numbers is √58?
Answer: 7 and 8
sqrt49 <sqrt58,sqrt64
Classify −1.75 as rational or irrational.
Write it as a fraction to support your answer.
Rational: −1.75 = -7/4
Rational numbers are numbers that can be turned into fractions.
Evaluate both expressions: √0.81 and −√0.81.
Answer: √0.81 = 0.9; −√0.81 = −0.9.
Evaluate ∛(−125).
Answer: −5
(−5)³ = −125
The decimal 0.414141… repeats the digits 41 forever. Is it rational or irrational? Explain.
Answer: Rational.
A repeating decimal can be written as a fraction; this number equals 41/99
On a number line, which point best represents √18?
A. 3.8 B. 4.2 C. 4.8 D. 5.2.
Answer: B. 4.2.
√18 is between sqrt4 4 and sqrt5 5
Select ALL numbers that are integers:
−√81, √20, 0, 7/2, ∛(−125).
−√81, 0, and ∛(−125)
Integers: whole numbers and their opposites
−√81 = −9
∛(−125) = −5
Solve x² = 196. Give every real solution.
Answer: x = 14 or x = −14.
Both 14² and (−14)² equal 196.
A cube-shaped storage box has a volume of 64 cubic inches. What is its edge length?
A. 4 in. B. 8 in.
C. 16 in. D. 32 in.
Answer: A. 4 inches
4³ = 64.
A student says √0.49 is irrational because 0.49 is a decimal. Is the student correct? Explain.
Answer: The student is not correct. √0.49 = 0.7 which means the value can be turned into a decimal: 7/10
A square garden has a perimeter of 30 meters. Find its side length and area.
Answer: Side = 30, Area = 56.25 sq meters.
Side:30 ÷ 4 = 7.5 meters.
Area: s^2 = 7.5² = 56.25 sq meters.
A student says every rational number is an integer.
Is the claim true?
Give a counterexample and explain the relationship between the sets.
False not all rational numbers are integers.
For example, 3/5 is rational but not an integer.
Every integer is rational because it can be written with denominator 1.
A square tile has an area of 12.25 square feet. Find its side length and perimeter.
Answer: Side length = 3.5 ft. Perimeter = 14 ft
Side length = √12.25 = 3.5 ft.
Perimeter = 3.5+3.5+3.5+3.5 = 4(3.5) = 14 ft.
A cube-shaped display has a volume of 15.625 cubic feet. Find its edge length and the area of one square face.
Answer: Edge length = 2.5 feet One face 6.25 sq ft.
Edge length = 2.5 feet because 2.5³ = 15.625.
One face has area 2.5² = 6.25 square feet
A calculator displays √7 as 2.645751311. A student says this proves √7 is a terminating decimal and therefore rational. Explain the mistake.
Answer: The display is a rounded approximation, not the complete exact decimal. √7 is irrational because 7 is not a perfect square; its exact decimal does not terminate or repeat.
Sort each number into the most specific group: integer, rational but not integer, or irrational.
√144, −11/4, √45, ∛(−64), 0.272727…
Integers: √144 = 12 and ∛(−64) = −4.
Rational but not integer: -11/4 and 0.272727… = 3/11.
Irrational: √45 because 45 is not a perfect square.
Maya writes √64 = ±8. Leo says x² = 64 has only the solution 8. Identify and correct each mistake.
Answer:
Maya: √64 means the principal root, so √64 = 8.
Leo: x² = 64 has two solutions, 8 and −8, because both square to 64.
Cube A has volume 8 cubic centimeters.
Cube B has volume 64 cubic centimeters.
A student claims Cube B’s edge is 8 times as long as Cube A’s because its volume is 8 times as large. Correct the claim.
Answer: Cube B’s edge is 2 times as long, not 8 times.
Cube A has edge ∛8 = 2 cm;
Cube B has edge ∛64 = 4 cm.
(Cube B)/(Cube A) = 4/2 =2
Without using a calculator, select ALL irrational numbers and justify each classification:
√121, √30, −√50, ∛(−27)
Irrational: √30 and −√50
Since sqrt30 and sqrt50 are not perfect squares; the decimal is irrational because it neither terminates nor repeats.
Rational: √121 = 11 and ∛(−27) = −3.
A square dog park has an area of 85 square meters. Estimate its perimeter to the nearest tenth of a meter. Then determine whether 37 meters of fencing is enough to enclose it completely (no gate opening).
Use your estimate to explain.
Answer: Perimeter= 36.9 m
Side = √85 ≈ 9.2195 m.
Perimeter = 4 x √85 ≈ 36.8782 m, which rounds to 36.9 m.
Yes, 37 m is enough; it more than the perimeter by about 0.12 m.