Laws of Exponents & Rational Exponents (Grade 10) — Worksheet & Memo

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Grade 10 · Exponents

Laws of Exponents & Rational Exponents (Grade 10)

MARKING GUIDELINE

Marks

Duration

55 minutes

Questions

Name:

Class:

Date:

Mark

/ 35

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
2⁻³ = …
(1)
A)18✓
B)−8
C)−6
D)−18
Answer: A — A negative exponent means the RECIPROCAL: 2⁻³ = 1 ÷ 2³ = 1/8.
- B — made the answer negative; a negative exponent flips the base instead
- C — multiplied 2 by −3
- D — the base is positive, so the answer stays positive
1.2
16^½ = …
(1)
A)4✓
B)8
C)32
D)116
Answer: A — A power of ½ is the SQUARE ROOT, and √16 = 4.
- B — halved 16 instead of square-rooting it
- C — multiplied 16 by 2
- D — treated the ½ as a negative exponent
1.3
8^2/3 = …
(1)
A)4✓
B)16
C)163
D)512
Answer: A — The denominator 3 is the CUBE root (∛8 = 2) and the numerator 2 is the power: 2² = 4.
- B — squared 8 and forgot the cube root
- C — multiplied 8 by 2/3 — an exponent is not a multiplier
- D — cubed 8 instead of taking its CUBE ROOT
1.4
(3x²)³ = …
(1)
A)27x⁶✓
B)9x⁶
C)3x⁶
D)27x⁵
Answer: A — Everything inside the bracket is cubed: 3³ = 27 and (x²)³ = x⁶.
- B — squared the 3 instead of cubing it
- C — never raised the 3 to the power at all
- D — ADDED the exponents 2 and 3 instead of multiplying them
1.5
If 3^x = 81, then x = …
(1)
A)4✓
B)27
C)3
D)5
Answer: A — 81 = 3⁴, so matching the bases gives x = 4.
- B — divided 81 by 3 instead of matching the bases
- C — read off the base instead of the exponent
- D — counted 3, 9, 27, 81, 243 and stopped one step too late
1.6
If 2^{x + 1} = 32, then x = …
(1)
A)4✓
B)5
C)15
D)16
Answer: A — 32 = 2⁵, so x + 1 = 5 and x = 4.
- B — solved x + 1 = 5 but forgot to subtract the 1
- C — divided 32 by 2 instead of matching the bases
- D — halved 32
1.7
x⁵ × x³x² = …
(1)
A)x⁶✓
B)x¹⁵
C)x⁴
D)x¹⁰
Answer: A — Multiplying ADDS exponents (5 + 3 = 8) and dividing SUBTRACTS them (8 − 2 = 6).
- B — multiplied the exponents 5 and 3 instead of adding them
- C — divided 8 by 2 instead of subtracting the 2
- D — added all three exponents instead of subtracting the last
1.8
∛(x⁵) written with a rational exponent is …
(1)
A)x^5/3✓
B)x^3/5
C)x¹⁵
D)x²
Answer: A — The ROOT becomes the denominator of the exponent and the power becomes the numerator.
- B — put the root on top and the power underneath — they are the other way round
- C — multiplied 5 by 3 instead of writing a fraction
- D — subtracted 3 from 5
1.9
5⁰ + 3⁻¹ = …
(1)
A)43✓
B)0
C)13
D)115
Answer: A — Anything to the power 0 is 1, and 3⁻¹ = 1/3, so the total is 1 + 1/3 = 4/3.
- B — took 5⁰ as 0 instead of 1, and 3⁻¹ as 0 too
- C — took 5⁰ as 0 and kept only the 1/3
- D — read 5⁰ as 1/5 and then MULTIPLIED instead of adding
1.10
A colony starts with 3 bacteria and DOUBLES every hour. After 6 hours there are …
(1)
A)192✓
B)36
C)64
D)18
Answer: A — Doubling six times multiplies by 2⁶ = 64, and 3 × 64 = 192.
- B — worked out 3 × 2 × 6 instead of 3 × 2⁶
- C — forgot the starting number of 3
- D — multiplied 3 by 6 — doubling is repeated multiplication, not addition

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
3^{x + 1} − 3^x3^x
(4)
Factorise the numerator: 3^x(3 − 1) (2)
= 2 × 3^x3^x (1) = 2 (1)
2.2
2^{x + 3}2^x
(3)
= 2^{(x + 3) − x} (2) = 2³ = 8 (1)
2.3
12a⁵b²4a²b⁶
(4)
12 ÷ 4 = 3 (1)
a⁵⁻² = a³ (1)
b²⁻⁶ = b⁻⁴ (1)
= 3a³b⁴ (1)

Question 3

[14 MARKS]

Calculate the following WITHOUT a calculator, and solve the equations that follow.

3.1
16^−3/4
(4)
Deal with the negative first: 116^3/4 (1)
16^3/4 = 2³ = 8 (2)
= 18 (1)
3.2
27^2/3
(3)
= (27)² — the CUBE root first (2) = 3² = 9 (1)
3.3
Solve for x: 5^2x = 125
(3)
125 = 5³ (1)
2x = 3 (1)
x = 1,5 (1)
3.4
Solve for x: 3^{3x − 1} = 27
(4)
27 = 3³ (1)
3x − 1 = 3 (2)
x = 43 (1)

TOTAL: 35 marks

This question paper consists of 3 questions.