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Neutrino properties such as the Majorana nature and the masses, which go beyond the standard model, are derived from the experimental double-beta decay (DBD) rate by using the DBD nuclear matrix element (NME). Theoretical evaluations for the NME, however, are very difficult. Single-charge exchange reactions (SCERs) and double-charge exchange reactions (DCERs) are used to study nuclear isospin (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>) and spin (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>) correlations involved in the DBD NME and to theoretically calculate the DBD NME. Single and double <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mi>σ</mi></mrow></semantics></math></inline-formula> NMEs for quasi-particle states are studied by SCERs and DCER. They are found to be reduced with respect to the quasi-particle model NMEs due to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mi>σ</mi></mrow></semantics></math></inline-formula> correlations. The impact of the SCER- and DCER-NMEs on the DBD NME is discussed.