化简 (75\sqrt[3]{7}+\sqrt[4]{3})/((\sqrt[3]{7)-\sqrt[4]{3})^2}

解答

化简

\frac{75 3 7 + 4 3}{( 3 7 - 4 3 ) ^{2}}

\frac{75 3 7 + 4 3}{3 7 ^{2} - 2 3 7 4 3 + 3}

十进制

406.42973 \dots

求解步骤

\frac{75 3 7 + 4 3}{( 3 7 - 4 3 ) ^{2}}

使用完全平方公式:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

(37 - 43)^{2} = (37)^{2} - 23743 + (43)^{2}

= \frac{75 3 7 + 4 3}{( 3 7 ) ^{2} - 2 3 7 4 3 + ( 4 3 ) ^{2}}

(37)^{2} - 23743 + (43)^{2} = 7^{\frac{2}{3}} - 23743 + 3

= \frac{75 3 7 + 4 3}{7 ^{\frac{2}{3}} - 2 3 7 4 3 + 3}

7^{\frac{2}{3}} = 3 7^{2}

= \frac{75 3 7 + 4 3}{3 7 ^{2} - 2 3 7 4 3 + 3}

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