The third proportional of \(\rm \frac{x^{2.5}}{y^{1.5}}\times z^{3.5}\ and \ x^{0.25}\times \frac{y^{0.25}}{z^{-0.25}}\) is ______

Given:

First term (a) = \(\rm \frac{x^{2.5}}{y^{1.5}}\times z^{3.5}\)

Second term (b) = \(\rm x^{0.25}\times \frac{y^{0.25}}{z^{-0.25}}\)

Formula Used:

If a, b, and c are in proportional, then a : b : : b : c or b² = ac

Therefore, the third proportional c = b²/a

Calculations:

⇒ b² = \(\rm (x^{0.25}\times y^{0.25}\times z^{0.25})^2\)

⇒ b² = \(\rm x^{0.5}\times y^{0.5}\times z^{0.5}\)

Now, the third proportional (c) = b²/a

⇒ c = \(\rm \frac{x^{0.5}\times y^{0.5}\times z^{0.5}}{\frac{x^{2.5}}{y^{1.5}}\times z^{3.5}}\)

⇒ c = \(\rm x^{(0.5-2.5)}\times y^{(0.5-(-1.5))}\times z^{(0.5-3.5)}\)

⇒ c = \(\rm x^{-2}\times y^{2}\times z^{-3}\)

⇒ c = \(\rm \frac{y^2}{x^2z^3}\)

∴ The correct answer is Option (3).