If $ x = 3^2 \times 2^3 $, then $ x^4 = ? $ | Mathematics MCQ

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If $x = 3^2 \times 2^3$ , then $x^4 =?$

$3^8 \times 2^3$

$3^6 \times 2^3$

$3^6 \times 2^7$

$3^8 \times 2^{12}$

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option D

$3^8 \times 2^{12}$

Quick Summary:

Apply the power rule $(a^m \times b^n)^p = a^{m \times p} \times b^{n \times p}$ directly to the expression.

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📐MAMath SolutionDirect Formula

📋 Given

x = 3² * 2³

🔢 Formula Used

(am)n=am×n(a^m)^n = a^{m \times n}(am)n=am×n

⚡ Exam Hall Shortcut / Speed Trick

Apply the power rule $(a^m \times b^n)^p = a^{m \times p} \times b^{n \times p}$ directly to the expression.

⚠️ Common Student Trap / Pitfall

Multiplying the exponents incorrectly, such as calculating $3^{2+4} \times 2^{3+4}$ instead of multiplying the power indices.

🔢 Step-by-Step Solution

Identify given expression

We are given $x = 3^2 \times 2^3$ and we need to find $x^4$ .

x=32×23x = 3^2 \times 2^3x=32×23

Apply power of a power rule

Substitute $x$ into the expression $x^4$ : $(3^2 \times 2^3)^4$ . Using the rule $(a \times b)^n = a^n \times b^n$ , we get $3^{2 \times 4} \times 2^{3 \times 4}$ .

(32×23)4=(32)4×(23)4(3^2 \times 2^3)^4 = (3^2)^4 \times (2^3)^4(32×23)4=(32)4×(23)4

Calculate exponents

Using $(a^m)^n = a^{m \times n}$ , we get $2 \times 4 = 8$ for base 3 and $3 \times 4 = 12$ for base 2.

38×2123^8 \times 2^{12}38×212

✅

D is correct because raising the given expression $x = 3^2 \times 2^3$ to the power of 4 results in $(3^2)^4 \times (2^3)^4 = 3^8 \times 2^{12}$ .

Core Concepts Used

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Exponent Laws Power of a Power Rule Algebraic Simplification

💡 EXAM TIP

This concept of power distribution is fundamental in simplifying complex indices in both Algebra and Physics (e.g., unit conversions involving square or cubic units).

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