115 Какие одночлены надо подставить вместо А и В, чтобы равенство превратилось в тождество? a) A⁵B⁷ = 32p¹⁰r²⁰q²¹s⁷; б) A⁵B¹² = 27x⁷y³z⁵t⁶.

• a) $$A^5B^7 = 32p^{10}r^{20}q^{21}s^7$$ $$A = (A^5)^{\frac{1}{5}} = (32p^{10}r^{20}q^{21}s^7)^{\frac{1}{5}} = 2p^2r^4q^{\frac{21}{5}}s^{\frac{7}{5}}$$ $$B = (B^7)^{\frac{1}{7}} = (32p^{10}r^{20}q^{21}s^7)^{\frac{1}{7}} = 2^{\frac{5}{7}}p^{\frac{10}{7}}r^{\frac{20}{7}}q^3s$$
• б) $$A^5B^{12} = 27x^7y^3z^5t^6$$ $$A = (A^5)^{\frac{1}{5}} = (27x^7y^3z^5t^6)^{\frac{1}{5}} = 3^{\frac{3}{5}}x^{\frac{7}{5}}y^{\frac{3}{5}}z t^{\frac{6}{5}}$$ $$B = (B^{12})^{\frac{1}{12}} = (27x^7y^3z^5t^6)^{\frac{1}{12}} = 3^{\frac{1}{4}}x^{\frac{7}{12}}y^{\frac{1}{4}}z^{\frac{5}{12}}t^{\frac{1}{2}}$$

Ответ: a) $$A = 2p^2r^4q^{\frac{21}{5}}s^{\frac{7}{5}}$$, $$B = 2^{\frac{5}{7}}p^{\frac{10}{7}}r^{\frac{20}{7}}q^3s$$; б) $$A = 3^{\frac{3}{5}}x^{\frac{7}{5}}y^{\frac{3}{5}}z t^{\frac{6}{5}}$$, $$B = 3^{\frac{1}{4}}x^{\frac{7}{12}}y^{\frac{1}{4}}z^{\frac{5}{12}}t^{\frac{1}{2}}$$