What will come in place of both the question marks in the following question? $\frac{$x$^{\frac{2}{3}}}{42} = \frac{5}{$x$^{\frac{1}{3}}}$

### Concept & Formula This problem relies on cross-multiplication for fractional equations and the fundamental rule of multiplying exponents with the same base. $$ a^m \times a^n = a^{m+n} $$ ### Step-by-Step Solution * **Given:** The equation is $\frac{$x$^{\frac{2}{3}}}{42} = \frac{5}{$x$^{\frac{1}{3}}}$. * **Calculation:** 1. Perform cross-multiplication to isolate the variable terms on one side and the constant terms on the other side: $x^{\frac{2}{3}} \times x^{\frac{1}{3}} = 42 \times 5$ 2. Apply the product rule for exponents on the left side by adding the powers: $\frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1$ 3. The left side simplifies to $x^1$, which is simply $x$. $x = 42 \times 5$ 4. Calculate the final product on the right side: $x = 210$ ### Exam Strategy & Shortcut Train yourself to spot complementary fractions. The powers $\frac{2}{3}$ and $\frac{1}{3}$ instantly add up to exactly $1$. Recognizing this means you do not need to write out intermediate steps. You can immediately see that cross-multiplying will yield $x^1 = 42 \times 5$. You can mentally calculate $40 \times 5 = 200$ and $2 \times 5 = 10$, summing to $210$. ### Common Pitfall Students sometimes get intimidated by fractional powers and attempt to cube both sides of the equation prematurely. While cubing both sides works, it creates unnecessarily large numbers ($42^3$ and $5^3$) that waste significant time and increase the risk of arithmetic errors. ### Final Answer **Therefore, the correct answer is 210.**