$3^{3.5} \times 21^2 \times 42^{2.5} \div 2^{2.5} \times 7^{3.5} = 21^x$

### Concept & Strategy When faced with multiple different bases, look for relationships that allow you to combine them into the target base (in this case, 21). Use the power of a quotient and power of a product rules. $$ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $$ $$ a^n \times b^n = (a \times b)^n $$ ### Step-by-Step Solution * The target base on the right-hand side is $21$. Our goal is to manipulate the left side to exclusively feature base $21$. * First, address the division part of the expression: $42^{2.5} \div 2^{2.5}$. * Since the exponents are the same, divide the bases: $\left(\frac{42}{2}\right)^{2.5} = 21^{2.5}$. * Next, look at the terms $3^{3.5}$ and $7^{3.5}$. They also share the same exponent. * Combine them by multiplying the bases: $(3 \times 7)^{3.5} = 21^{3.5}$. * Now, substitute these simplified terms back into the original equation. * $21^{3.5} \times 21^2 \times 21^{2.5} = 21^x$ * Since all bases are now identical ($21$) and are being multiplied, simply add their exponents. * $3.5 + 2 + 2.5 = 8$ * Therefore, $21^8 = 21^x$, which means $x = 8$. ### Exam Strategy & Shortcut Scan the equation and immediately identify the target base: $21$. Look for pairs that form 21. Mentally pair $42 \div 2$ to get $21^{2.5}$, and $3 \times 7$ to get $21^{3.5}$. You are left with exponents $3.5$, $2$, and $2.5$. Add them up in your head: $3.5 + 2.5 = 6$, plus $2 = 8$. Done in 10 seconds without writing a single step. ### Common Pitfall A major trap is blindly following BODMAS from left to right without looking for grouping opportunities, or incorrectly trying to apply laws of indices to terms with different bases and different exponents. Always unify the bases first. ### Final Answer **Therefore, the correct answer is 8.**