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Question Number 206104 by cortano21 last updated on 07/Apr/24

$$\downharpoonleft \underset{―}{}$$

Answered by dimentri last updated on 07/Apr/24

$$7^{-\mathrm{a}}=3^{-\mathrm{b}}=2^{-\mathrm{c}}=5^{-\mathrm{d}}=\mathrm{m}$$$$7={\mathrm{m}}^{-\frac{1}{\mathrm{a}}},3={\mathrm{m}}^{-\frac{1}{\mathrm{b}}},2={\mathrm{m}}^{-\frac{1}{\mathrm{c}}},5^{-\frac{1}{\mathrm{d}}}$$$${210}^{\mathrm{y}}={\mathrm{m}}^{-\mathrm{y}(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}+\frac{1}{\mathrm{d}})}=\mathrm{m}$$$$\frac{1}{\mathrm{y}}=-(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}+\frac{1}{\mathrm{d}})$$$$\frac{1}{\mathrm{y}}+\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}+\frac{1}{\mathrm{d}}=0$$

Answered by MATHEMATICSAM last updated on 07/Apr/24

$${210}^y=7^{-a}=3^{-b}=2^{-c}=5^{-d}=k\left(\mathrm{Let}\right)$$$$\Rightarrow 210=k^{\frac{1}{y}}$$$$\Rightarrow 7=k^{-\frac{1}{a}}$$$$\Rightarrow 3=k^{-\frac{1}{b}}$$$$\Rightarrow 2=k^{-\frac{1}{c}}$$$$\Rightarrow 5=k^{-\frac{1}{d}}$$$$$$$$210=7\times 3\times 2\times 5$$$$\Rightarrow k^{\frac{1}{y}}=k^{-\frac{1}{a}}\times k^{-\frac{1}{b}}\times k^{-\frac{1}{c}}\times k^{-\frac{1}{d}}$$$$\Rightarrow k^{\frac{1}{y}}=k^{-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}-\frac{1}{d}}$$$$\Rightarrow \frac{1}{y}=-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}-\frac{1}{d}$$$$\Rightarrow \frac{1}{y}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=0\left(\mathbf{Ans}\right)$$

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