$2^{3.6} \times 4^{3.6} \times 4^{3.6} \times (32)^{2.3} = (32)^{x}$

### Concept & Formula When multiple distinct bases share the exact same exponent, apply the power of a product rule in reverse to combine them into a single base. $$ a^n \times b^n \times c^n = (a \times b \times c)^n $$ ### Step-by-Step Solution * Given: $2^{3.6} \times 4^{3.6} \times 4^{3.6} \times (32)^{2.3} = (32)^x$ * Observe the first three terms. They all share the identical exponent of $3.6$. * Apply the rule to combine the bases: * $(2 \times 4 \times 4)^{3.6} \times (32)^{2.3} = (32)^x$ * Calculate the new combined base: $2 \times 4 = 8$, and $8 \times 4 = 32$. * Substitute this back: * $(32)^{3.6} \times (32)^{2.3} = (32)^x$ * Now, the equation features identical bases. Add the exponents on the left side: * $(32)^{3.6 + 2.3} = (32)^x$ * $(32)^{5.9} = (32)^x$ * Equate the exponents: * $x = 5.9$ ### Exam Strategy & Shortcut Before immediately converting everything to prime base $2$, look for structural patterns. Spotting that $2 \times 4 \times 4 = 32$ and that they all share the $3.6$ exponent allows you to instantly collapse three terms into $(32)^{3.6}$. The problem then becomes a trivial mental addition: $3.6 + 2.3 = 5.9$. ### Common Pitfall The most common inefficient approach is converting all bases to $2$ (e.g., making the left side $2^{3.6} \times 2^{7.2} \times 2^{7.2}$). While mathematically sound, it involves larger decimals and takes significantly more time, increasing the risk of calculation errors. ### Final Answer **Therefore, the correct answer is 5.9.**