The Laws of Exponents: Complete Exam Guide — Worksheet & Memo

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Grade 9 · Exponents

The Laws of Exponents: Complete Exam Guide

MARKING GUIDELINE

Marks

Duration

50 minutes

Questions

Name:

Class:

Date:

Mark

/ 31

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
3⁴ × 3² = …
(1)
A)3⁶✓
B)3⁸
C)9⁶
D)3²
Answer: A — Same base, multiplying: ADD the exponents.
- B — multiplied the exponents
- C — the BASE must not be multiplied too
- D — subtracted the exponents
1.2
(2x³)⁴ = …
(1)
A)8x¹²
B)16x¹²✓
C)16x⁷
D)2x¹²
Answer: B — Everything inside is raised to the 4th: 2⁴ = 16 and (x³)⁴ = x¹².
- A — used 2³ instead of 2⁴
- C — added the exponents instead of multiplying
- D — forgot to raise the 2
1.3
15x⁷/3x² = …
(1)
A)5x^3,5
B)12x⁵
C)5x⁹
D)5x⁵✓
Answer: D — 15 ÷ 3 = 5 and x⁷ ÷ x² = x⁵.
- A — divided the exponents
- B — SUBTRACTED the coefficients instead of dividing
- C — ADDED the exponents
1.4
If x ≠ 0, then x⁰ + 2⁰ = …
(1)
A)0
B)1
C)3
D)2✓
Answer: D — Each term is 1, and 1 + 1 = 2.
- A — a zero exponent gives 1, not 0
- B — only one of the two terms was counted
- C — treated 2⁰ as 2
1.5
Which one of the following is EQUAL to 2³ × 4²?
(1)
A)8⁵
B)2⁶
C)2⁷✓
D)8⁶
Answer: C — 4² = (2²)² = 2⁴, so the product is 2^{3 + 4}.
- A — multiplied the bases as well
- B — used 4² = 2³
- D — multiplied the exponents
1.6
(a²b³)² = …
(1)
A)a⁴b⁶✓
B)2a²b³
C)a⁴b⁵
D)a²b⁶
Answer: A — Multiply each exponent by 2.
- B — multiplied the whole term by 2
- C — added 2 to the exponent of b instead of doubling it
- D — forgot to double the exponent of a
1.7
5x³ + 5x³ = …
(1)
A)10x³✓
B)25x⁶
C)5x⁶
D)10x⁶
Answer: A — These are LIKE terms, so the exponent does not change.
- B — multiplied the coefficients too
- C — forgot to add the coefficients
- D — added the exponents as if multiplying
1.8
(x³ × x⁵)/(x² × x⁴) = …
(1)
A)x⁴
B)x¹⁴
C)x⁰
D)x²✓
Answer: D — Top: x⁸. Bottom: x⁶. So x^{8 − 6} = x².
- A — used 8 − 4, ignoring the x² below
- B — added all four exponents
- C — the exponents do not cancel completely
1.9
If 2^x = 32, then x = …
(1)
A)16
B)5✓
C)4
D)6
Answer: B — 2⁵ = 32.
- A — halved 32
- C — 2⁴ = 16
- D — 2⁶ = 64
1.10
(−3)² × (−3)³ = …
(1)
A)3⁵
B)9⁵
C)(−3)⁵✓
D)(−3)⁶
Answer: C — Same base, so add the exponents: 2 + 3 = 5. (The value is −243.)
- A — the base is −3, not 3, and an odd power stays negative
- B — the base must not be squared as well
- D — multiplied the exponents

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]

Simplify the following. Leave your answers with positive exponents.

2.1
12a⁵b²/4a²b⁶
(4)
12 ÷ 4 = 3 (1)
a^{5 − 2} = a³ (1)
b^{2 − 6} = b⁻⁴ (1)
= 3a³/b⁴ (1)
2.2
(2x²y³)³ × 3x⁻⁴
(4)
(2x²y³)³ = 8x⁶y⁹ (2)
× 3x⁻⁴ = 24x^{6 − 4}y⁹ (1)
= 24x²y⁹ (1)
2.3
Calculate WITHOUT a calculator: (−3²)(3)⁻²
(3)
−3² = −9 (the exponent applies to the 3 only) (1)
(3)⁻² = 1/9 (1)
−9 × 1/9 = −1 (1)

Question 3

[10 MARKS]

Answer the following WITHOUT using a calculator.

3.1
Calculate: 4² − [6 ÷ (−2)]³
(4)
6 ÷ (−2) = −3 (1)
(−3)³ = −27 (1)
16 − (−27) (1)
= 43 (1)
3.2
Calculate: 12 × 7⁰ + (5y)⁰, where y ≠ 0
(3)
7⁰ = 1, so 12 × 7⁰ = 12 (1)
(5y)⁰ = 1 — the zero exponent applies to the WHOLE bracket, not just the y (1)
= 13 (1)
3.3
Show that 2⁻³ can be written as a common fraction, and give its value.
(3)
2⁻³ = 1/2³ (1)
= 1/8 (2)

TOTAL: 31 marks

This question paper consists of 3 questions.