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Question Number 139861 by mathdanisur last updated on 01/May/21

$$\sqrt[3]{2\mathit{z}+1}+\sqrt[5]{8\mathit{z}+4}=\sqrt{16}$$

Answered by mindispower last updated on 01/May/21

$$z=\frac{7}{2},solution$$$$over\mathbb{R}$$$$f\left(z\right)=\sqrt[3]{2z+1}+\sqrt[5]{8z+4}isincreasefunction$$$$\underset{z\to -\mathrm{\infty }}{\lim }f\left(z\right)=-\mathrm{\infty }$$$$\underset{z\to +\mathrm{\infty }}{\lim }f\left(z\right)=+\mathrm{\infty },f\left(\frac{7}{2}\right)=4$$$$S=\left\{\frac{7}{2}\right\}$$$$$$$$$$

Commented by mathdanisur last updated on 02/May/21

$$thankssir$$

Answered by MJS_new last updated on 02/May/21

$$t=2z+1\iff z=\frac{t-1}{2}$$$$t^{1/3}+4^{1/5}{t}^{1/5}=4$$$$t=u^{15}$$$$u^5+4^{1/5}{u}^3-4=0$$$$u^5+2^{2/5}{u}^3-4=0$$$$\mathrm{looks}\mathrm{a}\mathrm{lot}\mathrm{like}u=2^v$$$$2^{5v}+2^{2/5}{2}^{3v}=4$$$$2^{5v}+2^{3v+2/5}=4$$$$v=1/5\Rightarrow u=2^{1/5}\Rightarrow t=8\Rightarrow z=\frac{7}{2}$$

Commented by mathdanisur last updated on 02/May/21

$$thankyousir$$

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