Integers: Squares, Cubes, Roots & Mixed Operations — Worksheet & Memo

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Grade 8 · Integers

Integers: Squares, Cubes, Roots & Mixed Operations

MARKING GUIDELINE

Marks

Duration

50 minutes

Questions

Name:

Class:

Date:

Mark

/ 33

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
144 = …
(1)
A)72
B)12✓
C)14
D)1 728
Answer: B — 12 × 12 = 144.
- A — HALVED 144 instead of rooting it
- C — guessed from the digits
- D — that is 12 CUBED
1.2
−27 = …
(1)
A)−3✓
B)3
C)−9
D)non-real
Answer: A — (−3)(−3)(−3) = −27; an ODD root of a negative is a real negative number.
- B — lost the sign — 3³ is +27
- C — that is −27 ÷ 3
- D — only an EVEN root of a negative is non-real
1.3
−4 is …
(1)
A)−2
B)2
C)non-real✓
D)−16
Answer: C — No real number multiplied by itself gives a negative.
- A — (−2)² = +4, not −4
- B — 2² = +4
- D — that is −4 squared
1.4
(−5)² + (−2)³ = …
(1)
A)17✓
B)−33
C)33
D)−17
Answer: A — 25 + (−8) = 17.
- B — made both terms negative
- C — made both terms positive
- D — subtracted 25 from 8
1.5
64 − 64 = …
(1)
A)0
B)4✓
C)−4
D)12
Answer: B — √64 = 8 and ∛64 = 4, so 8 − 4 = 4.
- A — assumed the two roots are equal
- C — subtracted the wrong way round
- D — ADDED the two roots
1.6
Which one of the following is a perfect square AND a perfect cube?
(1)
A)16
B)36
C)64✓
D)81
Answer: C — 64 = 8² and 64 = 4³.
- A — a square only
- B — a square only
- D — a square only
1.7
Between which TWO consecutive integers does √50 lie?
(1)
A)6 and 7
B)7 and 8✓
C)8 and 9
D)24 and 26
Answer: B — 7² = 49 and 8² = 64, and 50 falls between them.
- A — 36 and 49 — 50 is bigger than 49
- C — 64 and 81 are both above 50
- D — halved 50 instead of rooting it
1.8
(−1)¹⁰⁰ = …
(1)
A)−1
B)1✓
C)100
D)−100
Answer: B — An EVEN power of −1 is +1, because the negatives pair off.
- A — that is an ODD power of −1
- C — multiplied −1 by 100
- D — multiplied and kept the sign
1.9
A cube has a volume of 125 cm³. The length of one side is …
(1)
A)25 cm
B)5 cm✓
C)41,67 cm
D)15 cm
Answer: B — ∛125 = 5, because 5 × 5 × 5 = 125.
- A — took the square root of 625 by mistake
- C — divided the volume by 3
- D — divided by 3 and rounded
1.10
−2⁴ and (−2)⁴ differ because …
(1)
A)they are actually equal
B)the brackets decide whether the MINUS is also raised to the power✓
C)one is a cube
D)brackets never matter
Answer: B — −2⁴ = −16 but (−2)⁴ = +16.
- A — −16 and +16 are not equal
- C — both are fourth powers
- D — here they change the sign of the answer

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[10 MARKS]

Calculate the following WITHOUT using a calculator.

2.1
(−5)² − −4²
(4)
(−5)² = 25 (1)
−4² = −16 (1)
25 − (−16) (1) = 41 (1)
2.2
−27 + 64
(3)
−27 = −3 (1)
64 = 8 (1)
−3 + 8 = 5 (1)
2.3
(−3)² − 4 × (−2)
(3)
= 9 − (−8) (2) = 17 (1)

Question 3

[13 MARKS]

Calculate the following, showing the order in which you worked.

3.1
(−20 + 8) ÷ (−3) − (−5)
(4)
Brackets: −20 + 8 = −12 (1)
(−12) ÷ (−3) = 4 (1)
4 − (−5) (1) = 9 (1)
3.2
36 + 2(7 − 3)²
(4)
36 = 6 (1)
(7 − 3)² = 16 (1)
6 + 2(16) (1) = 38 (1)
3.3
Explain the difference between (−3)² and −3².
(2)
In (−3)² the bracket means the NEGATIVE is squared too, giving 9 (1)
In −3² only the 3 is squared and the minus stays, giving −9 (1)
3.4
Explain why −25 has no answer in the real numbers, but −27 does.
(3)
No real number multiplied by ITSELF gives a negative (1)
But (−3) × (−3) × (−3) = −27 (1), so an odd root of a negative does exist (1)

TOTAL: 33 marks

This question paper consists of 3 questions.