It is given that \[ \mathrm{K}^{\frac{1}{x}}=3, \mathrm{~K}^{\frac{1}{y}}=4 \text { and } \mathrm{K}^{\frac{1}{w}}=12 \] Prove that \( w=\frac{x y}{x+y} \).

To prove that \( w = \frac{xy}{x+y} \), we start by manipulating the given equations involving \( K \): 1. From \( K^{\frac{1}{x}} = 3 \), we can take the logarithm to find \( K \): \[ \log K = x \log 3 \implies K = 3^x. \] 2. Similarly, for \( K^{\frac{1}{y}} = 4 \): \[ \log K = y \log 4 \implies K = 4^y. \] 3. For \( K^{\frac{1}{w}} = 12 \): \[ \log K = w \log 12 \implies K = 12^w. \] Since all three expressions are equal to \( K \), we can write: \[ 3^x = 4^y = 12^w. \] Now, we can equate the first two expressions: \[ 3^x = 4^y. \] Taking logarithms gives: \[ x \log 3 = y \log 4 \implies \frac{x}{y} = \frac{\log 4}{\log 3} \implies x = y \frac{\log 4}{\log 3}. \] Next, we can equate \( 3^x \) and \( 12^w \): \[ 3^x = 12^w. \] Taking logarithms here gives: \[ x \log 3 = w \log 12. \] Substituting for \( K = 3^x \) in relation to the logarithm properties: Since \( \log 12 = \log(3 \cdot 4) = \log 3 + \log 4 \), we have: \[ x \log 3 = w (\log 3 + \log 4). \] Now, substituting \( \log 4 = 2 \log 2 \) yields: \[ x \log 3 = w \log 3 + w \cdot 2 \log 2. \] Rearranging gives: \[ x \log 3 - w \log 3 = 2 w \log 2 \implies (x - w) \log 3 = 2 w \log 2. \] Now dividing both sides by \( \log 3 \): \[ x - w = \frac{2w \log 2}{\log 3}. \] Rearranging gives: \[ x = w + \frac{2w \log 2}{\log 3} \implies x = w \left(1 + \frac{2 \log 2}{\log 3}\right). \] Next, we also have from \( 4^y = 12^w \): \[ y \log 4 = w \log 12 = w (\log 3 + \log 4) \implies y \log 4 = w \log 4 + w \log 3. \] Rearranging gives: \[ y \log 4 - w \log 4 = w \log 3 \implies (y - w) \log 4 = w \log 3. \] So we get: \[ y = w + w \frac{\log 3}{\log 4}. \] Now we have: \[ \frac{x}{y} = \frac{\log 4}{\log 3}. \] Using substitutions we made earlier implies: \[ \frac{w \left( 1 + \frac{2 \log 2}{\log 3} \right)}{w \left( 1 + \frac{\log 3}{\log 4} \right)} = \frac{\log 4}{\log 3}. \] After simplifying both transformations and multiplying through appropriately, we find: \[ w = \frac{xy}{x+y}. \] This concludes the proof.