Find the simplified value of the given expression. \( \frac{(2... | Stage I (CBT I) MCQ | Examoogle

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Stage I (CBT I)RRB ALP CBT-I 11 Mar, 2026 Shift 1

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Find the simplified value of the given expression.

$\frac{(2°3 \times 8°2)^3 \div (4°5 \times 16°2)}{(32°2 \div 2°4)^2}$

$2°{-2}$

$2°2$

$2°3$

$2°{-3}$

Correct Answer

📚 GK • GK Exponent LawsStage I (CBT I)RRB ALP CBT-I 11 Mar, 2026 Shift 1

Option D

$2°{-3}$

Quick Summary:

Given: Expression to simplify: $\frac{(2°3 \times 8°2)^3 \div (4°5 \times 16°2)}{(32°2 \div 2°4)^2}$

📐MAMath SolutionExponent Laws

📋 Given

Expression to simplify: $\frac{(2°3 \times 8°2)^3 \div (4°5 \times 16°2)}{(32°2 \div 2°4)^2}$

🔢 Formula Used

am×an=am+n,(am)n=am⋅n,am÷an=am−na^m \times a^n = a^{m+n}, \quad (a^m)^n = a^{m \cdot n}, \quad a^m \div a^n = a^{m-n}am×an=am+n,(am)n=am⋅n,am÷an=am−n

⚡ Exam Hall Shortcut / Speed Trick

Convert all terms to base 2 immediately: Numerator becomes $(2°3 \cdot 2°6)^3 \div (2°{10} \cdot 2°8) = 2°{27} \div 2°{18} = 2°9$ . Denominator becomes $(2°{10} \div 2°4)^2 = (2°6)^2 = 2°{12}$ . Answer is $2°{9 - 12} = 2°{-3}$ .

⚠️ Common Student Trap / Pitfall

Mixing up exponent rules like adding powers during division instead of subtracting, or misinterpreting $(2°5)^2$ as $2°7$ instead of $2°{10}$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Convert all bases to prime base 2

Express each base ( $8, 4, 16, 32$ ) as a power of $2$ : $8 = 2°3$ , $4 = 2°2$ , $16 = 2°4$ , and $32 = 2°5$ .

8=2°3,4=2°2,16=2°4,32=2°58 = 2°3, \quad 4 = 2°2, \quad 16 = 2°4, \quad 32 = 2°58=2°3,4=2°2,16=2°4,32=2°5

Simplify the Numerator

Substitute powers of $2$ into the numerator $(2°3 \times 8°2)^3 \div (4°5 \times 16°2)$ :

Inner term 1: $2°3 \times (2°3)^2 = 2°3 \times 2°6 = 2°9$
Raised to power 3: $(2°9)^3 = 2°{27}$
Inner term 2: $(2°2)^5 \times (2°4)^2 = 2°{10} \times 2°8 = 2°{18}$
Divide: $2°{27} \div 2°{18} = 2°{27 - 18} = 2°9$

Numerator=2°9\text{Numerator} = 2°9Numerator=2°9

Simplify the Denominator

Substitute powers of $2$ into the denominator $(32°2 \div 2°4)^2$ :

$32°2 = (2°5)^2 = 2°{10}$
$2°{10} \div 2°4 = 2°{10 - 4} = 2°6$
Square the result: $(2°6)^2 = 2°{12}$

Denominator=2°12\text{Denominator} = 2°{12}Denominator=2°12

Divide Numerator by Denominator

Divide the simplified numerator by the simplified denominator using exponent division rule $a^m \div a^n = a^{m-n}$ :

$\frac{2°9}{2°{12}} = 2°{9 - 12} = 2°{-3}$

Final Answer=2°−3\text{Final Answer} = 2°{-3}Final Answer=2°−3

✅

D is correct because simplifying both the numerator and denominator into base 2 yields $2°9 \div 2°{12} = 2°{-3}$ .

Core Concepts Used

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Laws of Exponents Base Harmonization Algebraic Simplification

💡 EXAM TIP

Converting composite numbers to prime factors (like $2, 3, 5$ ) is extremely useful across surds and indices, logarithm problems, and finding HCF/LCM in Number Systems.

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