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Let <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>≠</mo><mn>1</mn></mrow></semantics></math></inline-formula> be a positive real number and let <inline-formula><math display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> be a non-constant rational function with algebraic coefficients. In this paper, in particular, we prove that the set of algebraic numbers of the form <inline-formula><math display="inline"><semantics><msup><mi>α</mi><mrow><mi>P</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></msup></semantics></math></inline-formula>, with <i>T</i> transcendental, is dense in some open interval of <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>.