Squares, Cubes & Roots of Fractions — Worksheet & Memo

∑ DANEMATHICS

FREE CAPS MATHS RESOURCES

Grade 8 · Common Fractions

Squares, Cubes & Roots of Fractions

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
(23)² = …
(1)
A)43
B)49✓
C)29
D)69
Answer: B — Square the top AND the bottom.
- A — squared only the numerator
- C — squared only the denominator
- D — multiplied the top by 3
1.2
2536 = …
(1)
A)518
B)56✓
C)256
D)125
Answer: B — Root the top and the bottom separately.
- A — halved the denominator instead of rooting it
- C — rooted only the denominator
- D — turned the answer upside down
1.3
827 = …
(1)
A)23✓
B)49
C)29
D)83
Answer: A — 2³ = 8 and 3³ = 27.
- B — that is (2/3)²
- C — cube-rooted only the numerator
- D — cube-rooted only the denominator
1.4
(113)² = …
(1)
A)119
B)179✓
C)223
D)123
Answer: B — Change to an improper fraction first: (4/3)² = 16/9 = 1 7/9.
- A — squared the whole number and the fraction separately
- C — DOUBLED instead of squaring
- D — added 1/3 twice
1.5
Which one of the following is BIGGER than 14?
(1)
A)(14)²
B)14✓
C)(14)³
D)15
Answer: B — √(1/4) = 1/2, and taking the root of a number between 0 and 1 moves it CLOSER to 1.
- A — 1/16, which is smaller
- C — 1/64, which is smaller
- D — 1/5 is smaller than 1/4
1.6
916 + (12)² = …
(1)
A)54
B)1✓
C)32
D)78
Answer: B — 3/4 + 1/4 = 1.
- A — added the roots without squaring the second term
- C — doubled the first term
- D — used 1/8 for the square
1.7
A square has an area of 49100 m². The length of one side is …
(1)
A)7100
B)710✓
C)4910
D)2450
Answer: B — √(49/100) = 7/10 m.
- A — rooted only the numerator
- C — rooted only the denominator
- D — halved the area
1.8
Squaring a fraction between 0 and 1 makes it …
(1)
A)bigger
B)smaller✓
C)stay the same
D)negative
Answer: B — (1/2)² = 1/4, which is less than 1/2.
- A — that happens only for numbers greater than 1
- C — only 0 and 1 stay the same
- D — squaring never gives a negative
1.9
(35)² × 259 = …
(1)
A)1
B)35✓
C)925
D)53
Answer: B — 9/25 × 5/3 = 45/75 = 3/5.
- A — assumed the two cancel completely
- C — forgot the second factor
- D — used only the second factor
1.10
Which one of the following equals 18?
(1)
A)(12)³✓
B)(14)²
C)14
D)(12)²
Answer: A — (1/2)³ = 1/8.
- B — 1/16
- C — 1/2
- D — 1/4

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

[11 MARKS]

Calculate the following WITHOUT using a calculator. Leave your answers in the simplest form.

2.1
12² + 23²
(4)
= 14 + 49 (2)
LCD = 36: 936 + 1636 (1)
= 2536 (1)
2.2
2536, and hence 827
(3)
Root the top and the bottom separately (1)
2536 = 56 (1)
827 = 23, since 2³ = 8 and 3³ = 27 (1)
2.3
4964 − 2764
(4)
= 78 − 34 (2)
= 78 − 68 (1) = 18 (1)

Question 3

[12 MARKS]

Answer the questions below.

3.1
Calculate 113².
(3)
Change to an improper fraction FIRST: 43 (1)
169 (1) = 179 (1)
3.2
Nomsa writes 113² = 119. Explain her mistake, and calculate the correct answer.
(3)
She squared the whole number and the fraction separately (1)
A mixed number must first be written as an improper fraction: 113 = 43 (1)
43² = 169 = 179 (1)
3.3
A square has an area of 49100 m². Calculate the length of its side, and then its perimeter.
(4)
Side = 49100 = 710 m (2)
Perimeter = 4 × 710 (1) = 2810 = 245 m (1)
3.4
Explain why 14 is BIGGER than 14.
(2)
14 = 12 (1), and taking the root of a number between 0 and 1 moves it CLOSER to 1 (1)

TOTAL: 33 marks

This question paper consists of 3 questions.