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This study takes a detailed look at various inequalities related to the Euclidean operator radius. It examines groups of <i>n</i>-tuple operators, studying how they add up and multiply together. It also uncovers a unique power inequality specific to the Euclidean operator radius. The research broadens its scope to analyze how <i>n</i>-tuple operators, when used as parts of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> operator matrices, illustrate inequalities connected to the Euclidean operator radius. By using the Euclidean numerical radius and Euclidean operator norm for <i>n</i>-tuple operators, the study introduces a range of new inequalities. These inequalities not only set limits for the addition, multiplication, and Euclidean numerical radius of <i>n</i>-tuple operators but also help in establishing inequalities for the Euclidean operator radius. This process involves carefully examining the Euclidean numerical radius inequalities of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> operator matrices with <i>n</i>-tuple operators. Additionally, a new inequality is derived, focusing specifically on the Euclidean operator norm of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> operator matrices. Throughout, the research keeps circling back to the idea of finding and understanding symmetries in linear operators and matrices. The paper highlights the significance of symmetry in mathematics and its impact on various mathematical areas.