Exponents: Notation, Squares, Cubes & Roots — Worksheet & Memo

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Grade 8 · Exponents

Exponents: Notation, Squares, Cubes & Roots

MARKING GUIDELINE

Marks

Duration

55 minutes

Questions

Name:

Class:

Date:

Mark

/ 36

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
In 5³, the number 3 is called the …
(1)
A)base
B)exponent✓
C)product
D)coefficient
Answer: B — The exponent says how many times the base is multiplied by itself.
- A — the base is the 5
- C — the product is the answer, 125
- D — there is no coefficient here
1.2
5³ = …
(1)
A)15
B)125✓
C)8
D)53
Answer: B — 5 × 5 × 5 = 125.
- A — multiplied the base by the exponent
- C — added the base and the exponent
- D — read the two digits side by side
1.3
2 × 2 × 2 × 5 × 5 written in exponential form is …
(1)
A)2³ × 5²✓
B)2² × 5³
C)6 × 10
D)10⁵
Answer: A — Three 2s and two 5s.
- B — counted the 2s and 5s the wrong way round
- C — multiplied instead of counting
- D — added the number of factors
1.4
81 = …
(1)
A)9✓
B)40,5
C)8
D)6 561
Answer: A — 9 × 9 = 81.
- B — HALVED 81 instead of rooting it
- C — guessed from the digits
- D — that is 81 SQUARED
1.5
Which one of the following numbers 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
1.6
64 = …
(1)
A)8
B)4✓
C)21,3
D)16
Answer: B — 4 × 4 × 4 = 64.
- A — that is the SQUARE root of 64
- C — divided 64 by 3
- D — that is 64 ÷ 4
1.7
Written in scientific notation, 125 000 is …
(1)
A)12,5 × 10⁴
B)1,25 × 10⁵✓
C)125 × 10³
D)1,25 × 10⁴
Answer: B — The first number must be between 1 and 10, and the comma moves 5 places.
- A — 12,5 is not between 1 and 10
- C — 125 is not between 1 and 10
- D — the comma moves 5 places, not 4
1.8
3² + 4² = …
(1)
A)49
B)25✓
C)14
D)7²
Answer: B — 9 + 16 = 25.
- A — added the bases first and then squared
- C — added the bases and the exponents
- D — squaring is not distributive over addition
1.9
A square has an area of 144 cm². Its PERIMETER is …
(1)
A)12 cm
B)48 cm✓
C)576 cm
D)36 cm
Answer: B — Side = √144 = 12 cm, so the perimeter is 4 × 12 = 48 cm.
- A — that is the SIDE, not the perimeter
- C — multiplied the AREA by 4
- D — used 3 sides instead of 4
1.10
Which one of the following is the LARGEST?
(1)
A)2⁵✓
B)5²
C)3³
D)4²
Answer: A — The four values are 32, 25, 27 and 16.
- B — 25
- C — 27
- D — 16

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

[9 MARKS]

Calculate the following WITHOUT using a calculator. Show all your working.

2.1
81 + 125
(3)
= 9 (1)
+ 5 (1)
= 14 (1)
2.2
5² − 3³ + 49
(3)
= 25 − 27 + 7 (2)
= 5 (1)
2.3
Between which two consecutive whole numbers does 50 lie? Give a reason for your answer.
(3)
7² = 49 and 8² = 64 (2)
49 < 50 < 64, so 50 lies between 7 and 8 (1)

Question 3

[9 MARKS]

The diagram below shows a cube. The volume of the cube is 125 cm³.

3.1
Determine the length of one edge of the cube.
(3)
edge = 125 (1)
= 5 cm (2)
3.2
Hence calculate the total surface area of the cube.
(3)
A cube has 6 square faces (1)
SA = 6 × 5² = 6 × 25 (1)
= 150 cm² (1)
3.3
A square tile has an area of 144 cm². Calculate the perimeter of the tile.
(3)
side = 144 = 12 cm (2)
Perimeter = 4 × 12 = 48 cm (1)

Question 4

[8 MARKS]

Answer the following.

4.1
Write 2 × 2 × 2 × 5 × 5 in exponential form.
(2)
= 2³ × 5² (2)
4.2
Hence calculate the value of the expression in the previous question.
(2)
= 8 × 25 (1)
= 200 (1)
4.3
Only ONE of the numbers 16 ; 36 ; 64 ; 81 is both a perfect square AND a perfect cube. Write down that number and justify your answer.
(4)
64 (1)
64 = 8², so it is a perfect square (1)
64 = 4³, so it is a perfect cube (2)

TOTAL: 36 marks

This question paper consists of 4 questions.