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Question Number 70069 by tw000001 last updated on 01/Oct/19

$$\underset{n=1}{\prod ^5}\frac{{(12n-2)}^4+{18}^2}{{(12n-8)}^4+{18}^2}$$$$=\frac{({10}^4+324)({22}^4+324)({34}^4+324)({46}^4+324)({58}^4+324)}{(4^4+324)({16}^4+324)({28}^4+324)({40}^4+324)({52}^4+324)}$$

Commented by MJS last updated on 30/Sep/19

$$\mathrm{just}\mathrm{calculate}\mathrm{it}$$$$373$$

Commented by MJS last updated on 30/Sep/19

$$\mathrm{I}\mathrm{do}\mathrm{not}\mathrm{see}\mathrm{a}\mathrm{procedure}\mathrm{to}\mathrm{solve}\mathrm{this}$$

Answered by tw000001 last updated on 01/Oct/19

$$\mathrm{Use}{a}^4+b^4=(a^2+2b^2-2ab)(a^2+2b^2+2ab),$$$$\mathrm{so}\mathrm{it}\mathrm{can}\mathrm{find}\mathrm{out}{n}^4+324=n^4+4\cdot 3^4=[n(n-6)+18][n(n+6)+18].$$$$\frac{(4\cdot 10+18)(10\cdot 16+18)\cdot \cdot \cdot (52\cdot 58+18)(58\cdot 64+18)}{(-2\cdot 4+18)(4\cdot 10+18)\cdot \cdot \cdot (52\cdot 58+18)}$$$$=\frac{18+58\cdot 64}{18-2\cdot 4}$$$$=373$$

Commented by MJS last updated on 01/Oct/19

$$\mathrm{please}\mathrm{show}\mathrm{in}\mathrm{easy}\mathrm{steps}\mathrm{how}\mathrm{to}\mathrm{transform}$$$$\frac{{(12n-2)}^4+{18}^2}{{(12n-8)}^4+{18}^2}\mathrm{into}\mathrm{your}\mathrm{solution},\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}$$$$\mathrm{think}\mathrm{this}\mathrm{is}\mathrm{easy}\mathrm{to}\mathrm{see}$$

Commented by mind is power last updated on 01/Oct/19

$$x^4+4y^4={(x^2+2y^2)}^2-4x^2{y}^2=(x^2+2y^2-2xy)(x^2+2y^2+2xy)$$$$forx=12n-2andy=3$$$$\Rightarrow {(12n-2)}^4+4.3^4=({(12n-2)}^2+2.9-2.3(12n-2))({(12n-2)}^2+2.9+6(12n-2))$$$$=(144n^2-120n+34)(144n^2+24n+10)$$$${(12n-8)}^4+4.3^4=({(12n-8)}^2+2.3^2-2.3(12n-8))({(12n-8)}^2+18+6(12n-8))$$$$=(144n^2+130-264n)(144n^2-120n+34)$$$$\mathrm{\Pi }\frac{{(12n-2)}^4+{18}^2}{{(12n-8)}^4+{18}^2}=\mathrm{\Pi }\frac{(144n^2+24n+10)(144n^2-120n+34)}{(144n^2-120n+34)(144n^2-264n+130)}$$$$=\underset{k=1}{\prod ^m}\frac{144n^2+24n+10}{144n^2-264n+130}=\frac{\underset{k=1}{\prod ^m}(144n^2+24n+10)}{(144-264+130)\underset{k=1}{\prod ^{m-1}}(144{(n+1)}^2-264(n+1)+130)}$$$$=\frac{\underset{k=1}{\prod ^m}(144n^2+24n+10)}{10\underset{k=1}{\prod ^m}(144n^2+288n+144-264n-264+130)}=\frac{\underset{k=1}{\prod ^m}(144n^2+24n+10)}{10.\underset{k=1}{\prod ^{m-1}}(144n^2+24n+10)}$$$$=\frac{144m^2+24m+10}{10}$$$$HopthishelpyousirMJS$$$$$$

Commented by MJS last updated on 01/Oct/19

$$\mathrm{thank}\mathrm{you}$$

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