The Laws of Exponents (Grade 8) — Worksheet & Memo

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Grade 8 · Exponents

The Laws of Exponents (Grade 8)

MARKING GUIDELINE

Marks

Duration

1 hour

Questions

Name:

Class:

Date:

Mark

/ 39

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
x⁵ × x³ = …
(1)
A)x¹⁵
B)2x⁸
C)x⁸✓
D)x²
Answer: C — When the bases are the same and you MULTIPLY, you ADD the exponents.
- A — multiplied the exponents instead of adding them
- B — the bases are not added
- D — subtracted the exponents
1.2
x¹⁰ ÷ x⁴ = …
(1)
A)x¹⁴
B)x⁴⁰
C)x^2,5
D)x⁶✓
Answer: D — When you DIVIDE, you SUBTRACT the exponents.
- A — added the exponents
- B — multiplied the exponents
- C — divided the exponents
1.3
(x³)⁴ = …
(1)
A)x¹²✓
B)x⁷
C)x⁸¹
D)4x³
Answer: A — A power raised to a power MULTIPLIES the exponents.
- B — added them
- C — raised 3 to the 4th power
- D — multiplied the base by the outer exponent
1.4
7⁰ = …
(1)
A)1✓
B)0
C)undefined
D)7
Answer: A — Any non-zero number to the power 0 is 1.
- B — that would be 0⁷
- C — only 0⁰ is problematic
- D — that is 7¹
1.5
(2x)³ = …
(1)
A)8x
B)6x³
C)8x³✓
D)2x³
Answer: C — EVERYTHING inside the bracket is cubed, including the 2.
- A — forgot to cube the x
- B — multiplied 2 by 3 instead of cubing it
- D — cubed only the x
1.6
(x⁶ × x²)/x³ = …
(1)
A)x⁴
B)x³⁶
C)x¹¹
D)x⁵✓
Answer: D — 6 + 2 − 3 = 5.
- A — multiplied 6 × 2 and divided by 3
- B — multiplied the exponents
- C — added all three exponents
1.7
3x² × 4x⁵ = …
(1)
A)12x¹⁰
B)7x¹⁰
C)12x⁷✓
D)7x⁷
Answer: C — Multiply the coefficients and ADD the exponents.
- A — multiplied the exponents
- B — added the coefficients and multiplied the exponents
- D — ADDED the coefficients
1.8
Which one of the following is EQUAL to 2⁶?
(1)
A)2³ + 2³
B)2² × 2⁴✓
C)(2²)²
D)2² × 2³
Answer: B — 2 + 4 = 6.
- A — adding powers does not multiply them — this gives 16
- C — that is 2⁴
- D — that is 2⁵
1.9
x⁴ + x⁴ = …
(1)
A)2x⁴✓
B)x¹⁶
C)x⁸
D)2x⁸
Answer: A — These are LIKE terms, so they are counted, not multiplied — the exponent does not change.
- B — multiplied the exponents
- C — that is x⁴ × x⁴
- D — did both errors at once
1.10
(3x²y)² = …
(1)
A)9x²y²
B)9x⁴y²✓
C)6x⁴y²
D)3x⁴y²
Answer: B — Square each factor: 3² = 9, (x²)² = x⁴ and y².
- A — forgot to square the exponent of x
- C — multiplied 3 by 2 instead of squaring it
- D — did not square the 3

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]

Simplify the following. Leave your answers in exponential form.

2.1
(3x²)³ × 2x⁴
(4)
= 27x⁶ × 2x⁴ (2)
= 54x^{6 + 4} (1)
= 54x¹⁰ (1)
2.2
16a⁶b⁵/2a²b²
(3)
16 ÷ 2 = 8 (1)
a^{6 − 2} = a⁴ and b^{5 − 2} = b³ (1)
= 8a⁴b³ (1)
2.3
(2x³)² + 3x⁶
(3)
(2x³)² = 4x⁶ (1)
4x⁶ + 3x⁶ are like terms (1)
= 7x⁶ (1)

Question 3

[10 MARKS]

Calculate the following WITHOUT using a calculator.

3.1
2⁴ × 2³ ÷ 2⁵
(3)
= 2^{4 + 3 − 5} (2)
= 2² = 4 (1)
3.2
A learner writes 3² + 3² = 3⁴. Show by calculation that the learner is wrong, and write down the correct answer.
(4)
3² + 3² = 9 + 9 = 18 (2)
3⁴ = 81, and 18 ≠ 81 (1)
You may only add EXPONENTS when you MULTIPLY powers; here the answer is 18 (1)
3.3
Solve for x: 2^x = 32
(3)
32 = 2⁵ (1)
2^x = 2⁵, so the bases are the same (1)
x = 5 (1)

Question 4

[9 MARKS]

Solve for x in each of the following.

4.1
(2^x × 2^{x + 1})/16 − 1 = 7
(5)
(2^x × 2^{x + 1})/16 = 8 (1)
2^{2x + 1} = 8 × 16 = 128 (2)
2^{2x + 1} = 2⁷ (1)
2x + 1 = 7, so x = 3 (1)
4.2
(3^{x + 2})/3 = 27
(4)
3^{x + 2 − 1} = 27 (1)
3^{x + 1} = 3³ (2)
x + 1 = 3, so x = 2 (1)

TOTAL: 39 marks

This question paper consists of 4 questions.