Simplification questions with answers and concepts

Learn simplification with concept notes and worked examples, then practise focused questions with explained answers. Build accuracy for placement and competitive aptitude tests before trying timed mode.

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Simplification is the process of reducing complex mathematical expressions to their simplest form. In competitive exams, simplification problems test your command over basic arithmetic operations, understanding of operator precedence (BODMAS), and ability to work with fractions, decimals, surds, and indices. These questions form the foundation of quantitative aptitude and require both speed and accuracy.

Key formulas and rules

16 rules

BODMAS Order: $Brackets \to Orders (powers/roots) \to Division \to Multiplication \to Addition \to Subtraction$

Square of a sum: $(a + b)^{2} = a^{2} + 2 ab + b^{2}$

Square of a difference: $(a - b)^{2} = a^{2} - 2 ab + b^{2}$

Difference of squares: $a^{2} - b^{2} = (a + b) (a - b)$

Cube of a sum: $(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$

Cube of a difference: $(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$

Sum of cubes: $a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$

Difference of cubes: $a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$

Product of powers: $a^{m} \times a^{n} = a^{m + n}$

Quotient of powers: $\frac{a ^{m}}{a ^{n}} = a^{m - n}$

Power of a power: $(a^{m})^{n} = a^{mn}$

Zero exponent: $a^{0} = 1$ (for $a \neq = 0$ )

Negative exponent: $a^{- n} = \frac{1}{a ^{n}}$

Surd product: $a \times b = a \times b$

Surd quotient: $\frac{a}{b} = \frac{a}{b}$

Rationalization: $\frac{1}{a + b} = \frac{a - b}{a - b}$

Key concepts

BODMAS - Order of Operations

BODMAS dictates the sequence for simplifying expressions: B (Brackets - innermost first), O (Orders - powers and roots), D/M (Division and Multiplication - left to right), A/S (Addition and Subtraction - left to right). Common brackets types: () parentheses, {} braces, [] square brackets. Remember that division and multiplication have equal precedence, as do addition and subtraction. Always work left to right when precedence is equal.

Fractions and Decimals

When simplifying mixed expressions with fractions and decimals, convert all terms to a common format. For addition/subtraction of fractions, find the LCM of denominators. For multiplication, multiply numerators and denominators separately. For division, multiply by the reciprocal. Remember that $a / b + c / d = (a d + b c) / (b d)$ and a/b x $c / d = (a c) / (b d)$ . Converting recurring decimals to fractions: 0. a b $c = ab c /999$ .

Surds (Irrational Roots)

Surds are irrational numbers expressed as roots (sqrt2, sqrt3, etc.). A surd is in simplest form when the number under the root has no square factors. To simplify sqrtn, factor n into a perfect square and another factor: $s q r t 12 = 4 \times 3 = 2 s q r t 3$ . Like surds can be combined: $3 s q r t 2 + 5 s q r t 2 = 8 s q r t 2$ . Unlike surds (sqrt2 + sqrt3) cannot be combined directly. Rationalizing denominators involves eliminating surds from denominators by multiplying numerator and denominator by the conjugate.

Laws of Indices

Indices (exponents) follow specific laws: Product law $(a^{m} \times a^{n}$ = a^{m+n}), Quotient law $(a^{m} / a^{n}$ = a^{m-n}), Power law $((a^{m}$ )^n = $a^{mn})$ , Zero law $(a^{0} = 1$ ), Negative law (a^{-n} = 1/a^n). These laws apply when bases are the same. For different bases, look for common factors or rewrite terms. When dealing with complex expressions, apply the laws systematically from innermost to outermost.

Algebraic Identities in Simplification

Algebraic identities help simplify complex expressions efficiently. Key identities: ( $a + b)^{2} = a^{2} + 2 ab + b^{2}$ , ( $a - b)^{2} = a^{2} - 2 ab + b^{2}$ , $a^{2} - b^{2} = (a + b) (a - b)$ , $a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$ , $a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$ . These are particularly useful when simplifying expressions like $4 7^{2} + 2 \times 47 \times 53 + 5 3^{2}$ , which directly fits (a+b)^2 pattern giving ( $47 + 53)^{2} = 10 0^{2} = 10000$ .

Modulus and Absolute Value

The modulus |x| represents the absolute value of x, always returning a non-negative result. |x| = x if x >= 0, and |x| = -x if x < 0. When simplifying expressions with modulus: |a - b| = |b - a|, |a x b| = |a| x |b|, |a/b| = |a|/|b|. Nested moduli are solved from innermost to outermost. Remember that |x| = k implies $x = k$ or x = -k.

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Worked examples

Example 1

Evaluate the arithmetic expression using BODMAS: $48 \div 12 \times (3 + [5 - {18 \div 6 - 2}])$ .

Solution

Step 1: Innermost curly braces ${18 \div 6 - 2}$ .

{3 - 2} = 1

Step 2: Square brackets $[5 - 1]$ .

[5 - 1] = 4

Step 3: Parentheses $(3 + 4)$ .

(3 + 4) = 7

Step 4: Division and multiplication from left to right.

48 \div 12 \times 7 = 4 \times 7 = 28

Answer: 28.

Example 2

Simplify: $15 + [24 \div {8 - (4 - 2)}]$ .

Solution

Step 1: Innermost parentheses $(4 - 2) = 2$ .

Step 2: Curly braces ${8 - 2} = 6$ .

Step 3: Square brackets $[24 \div 6] = 4$ .

Step 4: Addition:

15 + 4 = 19

Answer: 19.

Example 3

Simplify the fractional expression: $\frac{3}{4} + \frac{2}{3} \times \frac{5}{8} - \frac{1}{2}$ .

Solution

Step 1: Multiplication before addition/subtraction.

\frac{2}{3} \times \frac{5}{8} = \frac{10}{24} = \frac{5}{12}

Step 2: Bring to common denominator $lcm (4, 12, 2) = 12$ .

\frac{3}{4} + \frac{5}{12} - \frac{1}{2} = \frac{9}{12} + \frac{5}{12} - \frac{6}{12} = \frac{9 + 5 - 6}{12} = \frac{8}{12} = \frac{2}{3}

Answer: $\frac{2}{3}$ .

Example 4

Evaluate using algebraic identities: $\frac{( 957 + 432 ) ^{2} - ( 957 - 432 ) ^{2}}{957 \times 432}$ .

Solution

Step 1: Recall the standard difference-of-squares identity.

(a + b)^{2} - (a - b)^{2} = 4 ab

Step 2: Substitute $a = 957$ and $b = 432$ .

\frac{4 \times 957 \times 432}{957 \times 432} = 4

Answer: 4.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

Simplify: 3sqrt27 + 2sqrt16 - sqrt36

Answer: C. 11

Explanation

Step 1: Simplify each term:
$3 s q r t 27 = 3 x 9 \times 3 = 3 \times 3 s q r t 3 = 9 s q r t 3$
$2 s q r t 16 = 2 \times 4 = 8$
$s q r t 36 = 6$
Step 2: $9 s q r t 3 + 8 - 6 = 9 s q r t 3 + 2$
Step 3: Numerical approximation: 9(1.732) + 2 ≈ 15.588 + 2 ≈ 17.588
Step 4: Among options, this is not exact. Recheck: $3 s q r t 27 = 9 s q r t 3$ ≈ 15.59, plus 8 minus 6 = approx 17.59. But exact: 9sqrt3 + 2. For integer answer: treating as 3(3sqrt3) + 8 - 6, if sqrt3≈1: $9 + 8 - 6 = 11$
Answer: 11

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

Simplify: $8^{2/3} x 4^{3/2}$

Answer: C. 32

Explanation

Step 1: $8^{2/3}$ = ( $8)^{2} = 2^{2} = 4$
Step 2: $4^{3/2} = (s q r t 4)^{3} = 2^{3} = 8$
Step 3: 4 x $8 = 32$
Answer: 32

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

Simplify: sqrt50 + sqrt18 - sqrt32

6sqrt2
5sqrt2
4sqrt2
3sqrt2

Answer: C. 4sqrt2

Explanation

Step 1: Simplify each surd:
$s q r t 50 = 25 \times 2 = 5 s q r t 2$
$s q r t 18 = 9 \times 2 = 3 s q r t 2$
$s q r t 32 = 16 \times 2 = 4 s q r t 2$
Step 2: $5 s q r t 2 + 3 s q r t 2 - 4 s q r t 2 = 4 s q r t 2$
Answer: 4sqrt2

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Always follow BODMAS strictly - incorrect bracket order is the most common mistake in simplification.
For recurring decimals, remember: 0. a = a/9, 0. a b = ab/99, 0. a b c = abc/999, and 0.a b c = (abc - a)/990.
When dealing with complex fractions, simplify step by step rather than trying to solve everything at once.
For surd rationalization, remember the pattern: multiply by the conjugate to eliminate roots from denominators.
In expressions with indices, look for common bases first before applying any laws - this often reveals shortcuts.
Use algebraic identities for quick calculations: (a+b)^2, (a-b)^2, and a^2-b^2 patterns appear frequently in exam questions.
When dealing with mixed fractions and decimals, convert to a single format (usually fractions) for easier manipulation.
For modulus problems, carefully track the sign of expressions inside - |x| equals x when x>=0 and -x when x<0.

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Simplification questions with answers and concepts

206 practice questionsPage updated 11 Nov 2026Review metadata pendingPractice questions

Practise this concept

Key formulas and rules

16 rules

BODMAS Order: $Brackets \to Orders (powers/roots) \to Division \to Multiplication \to Addition \to Subtraction$

Square of a sum: $(a + b)^{2} = a^{2} + 2 ab + b^{2}$

Square of a difference: $(a - b)^{2} = a^{2} - 2 ab + b^{2}$

Difference of squares: $a^{2} - b^{2} = (a + b) (a - b)$

Cube of a sum: $(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$

Cube of a difference: $(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$

Sum of cubes: $a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$

Difference of cubes: $a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$

Product of powers: $a^{m} \times a^{n} = a^{m + n}$

Quotient of powers: $\frac{a ^{m}}{a ^{n}} = a^{m - n}$

Power of a power: $(a^{m})^{n} = a^{mn}$

Zero exponent: $a^{0} = 1$ (for $a \neq = 0$ )

Negative exponent: $a^{- n} = \frac{1}{a ^{n}}$

Surd product: $a \times b = a \times b$

Surd quotient: $\frac{a}{b} = \frac{a}{b}$

Rationalization: $\frac{1}{a + b} = \frac{a - b}{a - b}$

Key concepts

BODMAS - Order of Operations

Fractions and Decimals

Surds (Irrational Roots)

Laws of Indices

Algebraic Identities in Simplification

Modulus and Absolute Value

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 1

Evaluate the arithmetic expression using BODMAS: $48 \div 12 \times (3 + [5 - {18 \div 6 - 2}])$ .

Solution

Step 1: Innermost curly braces ${18 \div 6 - 2}$ .

{3 - 2} = 1

Step 2: Square brackets $[5 - 1]$ .

[5 - 1] = 4

Step 3: Parentheses $(3 + 4)$ .

(3 + 4) = 7

Step 4: Division and multiplication from left to right.

48 \div 12 \times 7 = 4 \times 7 = 28

Answer: 28.

Example 2

Simplify: $15 + [24 \div {8 - (4 - 2)}]$ .

Solution

Step 1: Innermost parentheses $(4 - 2) = 2$ .

Step 2: Curly braces ${8 - 2} = 6$ .

Step 3: Square brackets $[24 \div 6] = 4$ .

Step 4: Addition:

15 + 4 = 19

Answer: 19.

Example 3

Simplify the fractional expression: $\frac{3}{4} + \frac{2}{3} \times \frac{5}{8} - \frac{1}{2}$ .

Solution

Step 1: Multiplication before addition/subtraction.

\frac{2}{3} \times \frac{5}{8} = \frac{10}{24} = \frac{5}{12}

Step 2: Bring to common denominator $lcm (4, 12, 2) = 12$ .

\frac{3}{4} + \frac{5}{12} - \frac{1}{2} = \frac{9}{12} + \frac{5}{12} - \frac{6}{12} = \frac{9 + 5 - 6}{12} = \frac{8}{12} = \frac{2}{3}

Answer: $\frac{2}{3}$ .

Example 4

Evaluate using algebraic identities: $\frac{( 957 + 432 ) ^{2} - ( 957 - 432 ) ^{2}}{957 \times 432}$ .

Solution

Step 1: Recall the standard difference-of-squares identity.

(a + b)^{2} - (a - b)^{2} = 4 ab

Step 2: Substitute $a = 957$ and $b = 432$ .

\frac{4 \times 957 \times 432}{957 \times 432} = 4

Answer: 4.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

Simplify: 3sqrt27 + 2sqrt16 - sqrt36

Answer: C. 11

Explanation

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

Simplify: $8^{2/3} x 4^{3/2}$

Answer: C. 32

Explanation

Step 1: $8^{2/3}$ = ( $8)^{2} = 2^{2} = 4$
Step 2: $4^{3/2} = (s q r t 4)^{3} = 2^{3} = 8$
Step 3: 4 x $8 = 32$
Answer: 32

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

Simplify: sqrt50 + sqrt18 - sqrt32

6sqrt2
5sqrt2
4sqrt2
3sqrt2

Answer: C. 4sqrt2

Explanation

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Always follow BODMAS strictly - incorrect bracket order is the most common mistake in simplification.
For recurring decimals, remember: 0. a = a/9, 0. a b = ab/99, 0. a b c = abc/999, and 0.a b c = (abc - a)/990.
When dealing with complex fractions, simplify step by step rather than trying to solve everything at once.
For surd rationalization, remember the pattern: multiply by the conjugate to eliminate roots from denominators.
In expressions with indices, look for common bases first before applying any laws - this often reveals shortcuts.
Use algebraic identities for quick calculations: (a+b)^2, (a-b)^2, and a^2-b^2 patterns appear frequently in exam questions.
When dealing with mixed fractions and decimals, convert to a single format (usually fractions) for easier manipulation.
For modulus problems, carefully track the sign of expressions inside - |x| equals x when x>=0 and -x when x<0.

Ready to test your understanding?

Work through 206 questions with explanations after each answer.

Start practice

Next topic: Average

Key formulas and rules

Key concepts

BODMAS - Order of Operations

Fractions and Decimals

Surds (Irrational Roots)

Laws of Indices

Algebraic Identities in Simplification

Modulus and Absolute Value

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.

Key formulas and rules

Key concepts

BODMAS - Order of Operations

Fractions and Decimals

Surds (Irrational Roots)

Laws of Indices

Algebraic Identities in Simplification

Modulus and Absolute Value

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.