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Question Number 206473 by hardmath last updated on 15/Apr/24

$$\mathrm{If}{4}^{\mathbf{p}}=5$$$$\mathrm{Find}:2^{3\mathbf{p}}=?$$

Answered by A5T last updated on 15/Apr/24

$$2^{2p}=5\Rightarrow 2^{3p}=\sqrt{125}$$$$$$$$p=\frac{{log}_{2}5}{2}\Rightarrow 2^{3p}=2^{{log}_2\left(5^{\frac{3}{2}}\right)}=5^{\frac{3}{2}}=\sqrt{125}=5\sqrt{5}$$

Commented by hardmath last updated on 15/Apr/24

$$\mathrm{thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{professor}$$

Answered by Rasheed.Sindhi last updated on 15/Apr/24

$$2^{3\mathrm{p}}={\left(2^2\right)}^{\frac{3\mathrm{p}}{2}}={\left(4^{\mathrm{p}}\right)}^{3/2}=5^{3/2}=\sqrt{125}=5\sqrt{5}$$

Answered by BaliramKumar last updated on 16/Apr/24

$$2^{2\mathrm{p}}=5\Rightarrow 2=5^{\frac{1}{2\mathrm{p}}}$$$$2^{3\mathrm{p}}={\left(5^{\frac{1}{2\mathrm{p}}}\right)}^{3\mathrm{p}}=5^{\frac{3\mathrm{p}}{2\mathrm{p}}}=5^{\frac{3}{2}}=5\sqrt{5}$$

Answered by Skabetix last updated on 16/Apr/24

$${\left(2^2\right)}^p=5$$$$\iff 2^{2p}=5$$$$\iff {\left(2^{2p}\right)}^{\frac{3}{2}}=5^{\frac{3}{2}}$$$$\iff 2^{2p\times \frac{3}{2}}=5^{\frac{3}{2}}$$$$\iff 2^{3p}=5^{\frac{3}{2}}=5\sqrt{5}$$

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