For the positive real numbers m, n, x and y, m3x4 = n3y4. If m2 < n2, which of the following is true?

To analyze the given condition $m^3 x^4 = n^3 y^4$ under the constraint $m^2 < n^2$, we can follow the steps outlined below:

1. Break Down the Problem

We have two equations:

• $m^3 x^4 = n^3 y^4$
• $m^2 < n^2$

We need to determine the relationship between $x$ and $y$ or any other conclusions that can be drawn from these equations given the inequality.

2. Relevant Concepts

Using properties of inequalities and the relationships between the variables, we can manipulate the initial equation to isolate the variables of interest:

• From $m^3 x^4 = n^3 y^4$, we can express $\frac{x^4}{y^4}$ in terms of $\frac{m^3}{n^3}$.

3. Analysis and Detail

Rearranging the original equation: $\frac{x^4}{y^4} = \frac{n^3}{m^3}$ This can be expressed as: $\left( \frac{x}{y} \right)^4 = \frac{n^3}{m^3}$ Taking the fourth root of both sides gives: $\frac{x}{y} = \left(\frac{n^3}{m^3}\right)^{\frac{1}{4}} = \frac{n^{\frac{3}{4}}}{m^{\frac{3}{4}}}$

Because $m^2 < n^2$, we can deduce: $\frac{m}{n} < 1 \quad \Rightarrow \quad \left( \frac{m}{n} \right)^{\frac{3}{4}} < 1$ Thus, it implies: $\frac{n^{\frac{3}{4}}}{m^{\frac{3}{4}}} > 1 \quad \Rightarrow \quad \frac{x}{y} > 1 \quad \Rightarrow \quad x > y$

4. Verify and Summarize

From the deduction, $m^2 < n^2$ leads us to $\frac{n^{\frac{3}{4}}}{m^{\frac{3}{4}}} > 1$, resulting in $x > y$.

Final Answer

Thus, if $m^2 < n^2$, it follows that $x > y$.