If $ 2^{K} \times 4^{12} = 16^{11}$, then the value of K is: | Mathematics MCQ

MathematicsPrevious Year Paper 2024

If $2^{K} \times 4^{12} = 16^{11}$ , then the value of K is:

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option A

Quick Summary:

Express all terms with base 2: $4 = 2^2$ and $16 = 2^4$ . The equation becomes $2^K \times (2^2)^{12} = (2^4)^{11}$ , which simplifies to $K + 24 = 44$ .

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📐MAMath SolutionDirect Formula

📋 Given

The equation given is $2^K \times 4^{12} = 16^{11}$ .

🔢 Formula Used

(am)n=am×n,am×an=am+n(a^m)^n = a^{m \times n}, \quad a^m \times a^n = a^{m+n}(am)n=am×n,am×an=am+n

⚡ Exam Hall Shortcut / Speed Trick

Express all terms with base 2: $4 = 2^2$ and $16 = 2^4$ . The equation becomes $2^K \times (2^2)^{12} = (2^4)^{11}$ , which simplifies to $K + 24 = 44$ .

⚠️ Common Student Trap / Pitfall

Students often make mistakes by multiplying the bases or failing to convert 4 and 16 into powers of 2 before equating the exponents.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Express all bases as powers of 2

Since the base of the first term is 2, express $4$ as $2^2$ and $16$ as $2^4$ .

4=22,16=244 = 2^2, \quad 16 = 2^44=22,16=24

Substitute and simplify powers

Substitute these into the equation and use the power of a power rule $(a^m)^n = a^{mn}$ .

2K×(22)12=(24)11  ⟹  2K×224=2442^K \times (2^2)^{12} = (2^4)^{11} \implies 2^K \times 2^{24} = 2^{44}2K×(22)12=(24)11⟹2K×224=244

Apply product rule and equate exponents

Use $a^m \times a^n = a^{m+n}$ to simplify the left side, then equate the exponents of base 2.

2K+24=244  ⟹  K+24=442^{K+24} = 2^{44} \implies K + 24 = 442K+24=244⟹K+24=44

Solve for K

Subtract 24 from both sides to isolate K.

K=44−24=20K = 44 - 24 = 20K=44−24=20

✅

A is correct because the calculation leads to K = 20.

Core Concepts Used

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Laws of Exponents Base Conversion Algebraic Equations

💡 EXAM TIP

This exponent rule is fundamental for logarithmic equations and solving complex base-conversion problems in competitive exams.

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