Civil Service Exam (Professional) Numerical Ability — Algebra, Exponents & Number SeriesExam Answer Templates

Examiner tip

Always write the exponent law you're using - this shows you understand the concept, not just memorized calculations

Model answer

Given: 3^4 × 3^2

Using the Product Rule for exponents: a^m × a^n = a^(m+n)
3^4 × 3^2 = 3^(4+2) = 3^6 = 729

Therefore, 3^4 × 3^2 = 729

Answer structure

• Line 1: State the given expression [0.5 marks]
• Line 2: Apply the correct exponent law [1 mark]
• Line 3: Calculate the final answer [0.5 marks]

Scoring breakdown

Common mark deductions

• Not showing the rule used
• Arithmetic errors in final calculation
• Not stating the final answer clearly

Key phrases to include

• Product Rule
• a^m × a^n = a^(m+n)
• final answer

Examiner tip

Always show the pattern-finding process explicitly - don't just state the answer

Model answer

Given series: 2, 5, 8, 11, 14, ?

Step 1: Find the pattern
5 - 2 = 3
8 - 5 = 3
11 - 8 = 3
14 - 11 = 3

Step 2: Identify the sequence type
This is an arithmetic sequence with common difference d = 3

Step 3: Find the next term
Next term = 14 + 3 = 17

Therefore, the next term is 17.

Answer structure

• Step 1: Calculate differences between consecutive terms [1 mark]
• Step 2: Identify the type of sequence [1 mark]
• Step 3: Apply pattern to find next term [1 mark]

Scoring breakdown

Common mark deductions

• Not showing difference calculations
• Incorrect identification of sequence type
• Arithmetic errors

Key phrases to include

• arithmetic sequence
• common difference
• pattern

Examiner tip

Always verify your answer by substituting back into the original equation

Model answer

Given: 2^(x+1) = 16

Step 1: Express 16 as a power of 2
16 = 2^4

Step 2: Rewrite the equation
2^(x+1) = 2^4

Step 3: Since bases are equal, equate exponents
x + 1 = 4
x = 4 - 1
x = 3

Verification: 2^(3+1) = 2^4 = 16 ✓

Therefore, x = 3

Answer structure

• Step 1: Express the right side as power of same base [1 mark]
• Step 2: Apply the principle that equal bases mean equal exponents [1 mark]
• Step 3: Solve for the variable and verify [1 mark]

Scoring breakdown

Common mark deductions

• Not converting to same base
• Algebraic errors
• Missing verification step

Key phrases to include

• express as power
• equal bases
• verification

Examiner tip

Break down complex expressions into manageable parts - coefficients, then each variable separately

Model answer

Given: (8x^3y^6)/(4x^2y^4)

Step 1: Separate coefficients and variables
= (8/4) × (x^3/x^2) × (y^6/y^4)

Step 2: Apply quotient rule for exponents
= 2 × x^(3-2) × y^(6-4)
= 2 × x^1 × y^2
= 2xy^2

Therefore, (8x^3y^6)/(4x^2y^4) = 2xy^2

Answer structure

• Step 1: Separate numerical and variable parts [1 mark]
• Step 2: Apply quotient rule and simplify [1 mark]

Scoring breakdown

Common mark deductions

• Incorrect application of quotient rule
• Arithmetic errors with coefficients
• Not simplifying completely

Key phrases to include

• quotient rule
• a^m/a^n = a^(m-n)
• simplify

Examiner tip

Remember: any non-zero number to the power 0 always equals 1

Model answer

4^0 + 5^0 = 1 + 1 = 2
(Any non-zero number raised to power 0 equals 1)

Answer structure

• Apply zero exponent rule and calculate [1 mark]

Scoring breakdown

Common mark deductions

• Not knowing that a^0 = 1
• Arithmetic errors

Key phrases to include

• zero exponent rule
• equals 1

Examiner tip

Always verify your answer fits the pattern by checking it works with surrounding terms

Model answer

Given series: 3, 6, 12, ?, 48

Step 1: Find the common ratio
r = 6/3 = 2
r = 12/6 = 2

Step 2: Verify it's a geometric series
Common ratio r = 2 (constant)

Step 3: Find the missing term
Missing term = 12 × 2 = 24

Verification: 24 × 2 = 48 ✓

Therefore, the missing term is 24.

Answer structure

• Step 1: Calculate common ratio from given terms [1 mark]
• Step 2: Confirm geometric sequence pattern [1 mark]
• Step 3: Apply ratio to find missing term [1 mark]

Scoring breakdown

Common mark deductions

• Not identifying the common ratio
• Calculation errors
• Missing verification

Key phrases to include

• geometric series
• common ratio
• verification

Examiner tip

State the exponent law you're using - it shows conceptual understanding

Model answer

Given: (3^2)^3

Using Power Rule: (a^m)^n = a^(mn)
(3^2)^3 = 3^(2×3) = 3^6

Calculating: 3^6 = 729

Therefore, (3^2)^3 = 729

Answer structure

• Apply power rule for exponents [1 mark]
• Calculate final numerical value [1 mark]

Scoring breakdown

Common mark deductions

• Incorrect application of power rule
• Calculation errors
• Not showing the rule

Key phrases to include

• Power Rule
• (a^m)^n = a^(mn)

Examiner tip

Always express negative exponents as positive fractions unless specifically asked for decimal form

Model answer

Given: 2^(-3)

Using Negative Exponent Rule: a^(-n) = 1/a^n
2^(-3) = 1/2^3 = 1/8

Therefore, 2^(-3) = 1/8

Answer structure

• Apply negative exponent rule [1 mark]
• Calculate final fraction form [1 mark]

Scoring breakdown

Common mark deductions

• Not knowing negative exponent rule
• Arithmetic errors
• Leaving answer as decimal instead of fraction

Key phrases to include

• Negative Exponent Rule
• a^(-n) = 1/a^n

Examiner tip

Always state the formula you're using - it demonstrates method knowledge even if calculation errors occur

Model answer

Given: Arithmetic sequence 5, 9, 13, 17, ...
To find: 8th term (a₈)

Step 1: Identify first term and common difference
a₁ = 5
d = 9 - 5 = 4

Step 2: Use arithmetic sequence formula
aₙ = a₁ + (n-1)d
a₈ = 5 + (8-1)×4
a₈ = 5 + 7×4
a₈ = 5 + 28
a₈ = 33

Therefore, the 8th term is 33.

Answer structure

• Step 1: Identify first term and common difference [1 mark]
• Step 2: Apply arithmetic sequence formula [1 mark]
• Step 3: Calculate final answer [1 mark]

Scoring breakdown

Common mark deductions

• Wrong common difference calculation
• Formula errors
• Arithmetic mistakes

Key phrases to include

• arithmetic sequence
• common difference
• formula aₙ = a₁ + (n-1)d

Examiner tip

Memorize perfect cubes up to 10³ for quick recognition

Model answer

x³ = 27
x = ∛27 = 3

Therefore, x = 3

Answer structure

• Take cube root of both sides to find x = 3 [1 mark]

Scoring breakdown

Common mark deductions

• Not recognizing 27 as perfect cube
• Calculation errors

Key phrases to include

• cube root
• ∛27 = 3

Examiner tip

Look for common mathematical sequences like squares, cubes, or factorial patterns

Model answer

Given series: 1, 4, 9, 16, 25, ?, ?

Step 1: Analyze the pattern
1 = 1²
4 = 2²
9 = 3²
16 = 4²
25 = 5²

Step 2: Identify the sequence type
This is a perfect squares sequence: n²

Step 3: Find next two terms
6² = 36
7² = 49

Therefore, the next two terms are 36 and 49.

Answer structure

• Step 1: Express each term as perfect square [1 mark]
• Step 2: Identify pattern as n² sequence [1 mark]
• Step 3: Calculate next two terms [1 mark]

Scoring breakdown

Common mark deductions

• Not recognizing perfect square pattern
• Calculation errors
• Finding only one term instead of two

Key phrases to include

• perfect squares
• n² sequence
• pattern

Examiner tip

Remember that fractional exponents represent roots - the denominator tells you which root to take

Model answer

Given: 16^(1/4)

Using Fractional Exponent Rule: a^(1/n) = ⁿ√a
16^(1/4) = ⁴√16

Since 16 = 2⁴, we have:
⁴√16 = ⁴√(2⁴) = 2

Therefore, 16^(1/4) = 2

Answer structure

• Apply fractional exponent rule [1 mark]
• Evaluate the fourth root correctly [1 mark]

Scoring breakdown

Common mark deductions

• Not knowing fractional exponent rule
• Incorrect root calculation
• Not recognizing 16 as 2⁴

Key phrases to include

• Fractional Exponent Rule
• a^(1/n) = ⁿ√a
• fourth root

Examiner tip

Structure your answer with clear headings and always include a worked example to demonstrate application

Model answer

Steps to Simplify Exponential Expressions:

Step 1: Apply Product Rule
Multiply terms with same base by adding exponents: aᵐ × aⁿ = aᵐ⁺ⁿ

Step 2: Apply Quotient Rule
Divide terms with same base by subtracting exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Step 3: Apply Power Rule
Raise a power to another power by multiplying exponents: (aᵐ)ⁿ = aᵐⁿ

Step 4: Apply Zero Exponent Rule
Any non-zero number to power 0 equals 1: a⁰ = 1

Step 5: Apply Negative Exponent Rule
Convert negative exponents to positive: a⁻ⁿ = 1/aⁿ

Example: Simplify (2³ × 2⁵) ÷ 2⁴
Step 1: 2³ × 2⁵ = 2³⁺⁵ = 2⁸ (Product Rule)
Step 2: 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ (Quotient Rule)
Step 3: 2⁴ = 16

Therefore, the expression simplifies to 16.

Answer structure

• Step 1: Explain Product Rule with formula [1 mark]
• Step 2: Explain Quotient Rule with formula [1 mark]
• Step 3: Explain Power Rule with formula [1 mark]
• Step 4: Explain Zero and Negative Exponent Rules [1 mark]
• Step 5: Provide complete worked example [1 mark]

Scoring breakdown

Common mark deductions

• Missing any of the five rules
• Incorrect formulas
• Poor or missing example
• Not showing step-by-step solution

Key phrases to include

• Product Rule
• Quotient Rule
• Power Rule
• Zero Exponent Rule
• Negative Exponent Rule
• worked example

Examiner tip

Always verify your common ratio with at least two consecutive pairs before proceeding

Model answer

Given: Geometric series 2, 6, 18, 54, ...
To find: Sum of first 6 terms (S₆)

Step 1: Identify first term and common ratio
a₁ = 2
r = 6/2 = 3

Step 2: Verify common ratio
18/6 = 3 ✓
54/18 = 3 ✓

Step 3: Find the 5th and 6th terms
a₅ = 2 × 3⁴ = 2 × 81 = 162
a₆ = 2 × 3⁵ = 2 × 243 = 486

Step 4: Use geometric series sum formula
Sₙ = a₁(rⁿ - 1)/(r - 1)
S₆ = 2(3⁶ - 1)/(3 - 1)
S₆ = 2(729 - 1)/2
S₆ = 2(728)/2
S₆ = 728

Therefore, the sum of first 6 terms is 728.

Answer structure

• Step 1: Identify first term and common ratio [1 mark]
• Step 2: Verify the common ratio [1 mark]
• Step 3: Find remaining terms if needed [1 mark]
• Step 4: Apply geometric series sum formula [1 mark]
• Step 5: Calculate final answer correctly [1 mark]

Scoring breakdown

Common mark deductions

• Wrong common ratio
• Using wrong formula
• Calculation errors
• Not verifying the ratio

Key phrases to include

• geometric series
• common ratio
• sum formula Sₙ = a₁(rⁿ - 1)/(r - 1)
• verification

Examiner tip

For geometric sequences, always express the general term in its simplest form

Model answer

Given sequence: 1/2, 1/4, 1/8, 1/16, ...

Step 1: Find the pattern
(1/4)/(1/2) = 1/2
(1/8)/(1/4) = 1/2
(1/16)/(1/8) = 1/2

Step 2: Identify sequence type
This is a geometric sequence with:
a₁ = 1/2
r = 1/2

Step 3: Find general term
For geometric sequence: aₙ = a₁ × rⁿ⁻¹
aₙ = (1/2) × (1/2)ⁿ⁻¹
aₙ = (1/2)ⁿ
aₙ = 1/2ⁿ

Therefore, this is a geometric sequence with general term aₙ = 1/2ⁿ

Answer structure

• Step 1: Calculate ratios to find pattern [1 mark]
• Step 2: Identify as geometric sequence [1 mark]
• Step 3: Derive general term formula [1 mark]

Scoring breakdown

Common mark deductions

• Not identifying sequence type
• Wrong common ratio
• Incorrect general term formula

Key phrases to include

• geometric sequence
• common ratio
• general term
• aₙ = a₁ × rⁿ⁻¹

Do

• State the rule/formula if applicable
• Give a clear final answer
• Show one key step if needed

Don't

• Over-explain simple concepts
• Show unnecessary working
• Leave answer unclear

Do

• State what rule you're using
• Show the key calculation step
• Present final answer clearly

Don't

• Skip the method/rule
• Make careless arithmetic errors
• Forget to state the final answer

Do

• Break into logical steps
• Show all working clearly
• Verify answer when possible
• Label steps appropriately

Don't

• Jump steps without explanation
• Mix up different concepts
• Forget intermediate calculations

Do

• Use clear headings/structure
• Provide complete explanations
• Include worked examples
• Cover all required points
• Show mastery of topic

Don't

• Rush through explanations
• Miss key concepts
• Provide incomplete examples
• Write without structure