Menu Close

If-4-p-5-Find-2-3p-

Tinku Tara 0 Comments
Pin




Question Number 206473 by hardmath last updated on 15/Apr/24
If 4^p = 5 Find: 2^(3p) = ?
$$\mathrm{If}\:\:\:\mathrm{4}^{\boldsymbol{\mathrm{p}}} \:=\:\mathrm{5} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{2}^{\mathrm{3}\boldsymbol{\mathrm{p}}} \:=\:? \\ $$
Answered by A5T last updated on 15/Apr/24
2^(2p) =5⇒2^(3p) =(√(125)) p=((log_2 5)/2)⇒2^(3p) =2^(log_2 (5^(3/2) )) =5^(3/2) =(√(125))=5(√5)
$$\mathrm{2}^{\mathrm{2}{p}} =\mathrm{5}\Rightarrow\mathrm{2}^{\mathrm{3}{p}} =\sqrt{\mathrm{125}} \\ $$$$ \\ $$$${p}=\frac{{log}_{\mathrm{2}} \mathrm{5}}{\mathrm{2}}\Rightarrow\mathrm{2}^{\mathrm{3}{p}} =\mathrm{2}^{{log}_{\mathrm{2}} \left(\mathrm{5}^{\frac{\mathrm{3}}{\mathrm{2}}} \right)} =\mathrm{5}^{\frac{\mathrm{3}}{\mathrm{2}}} =\sqrt{\mathrm{125}}=\mathrm{5}\sqrt{\mathrm{5}} \\ $$
Commented by hardmath last updated on 15/Apr/24
thank you very much professor
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{professor}\: \\ $$
Answered by Rasheed.Sindhi last updated on 15/Apr/24
2^(3p) =(2^2 )^((3p)/2) =(4^p )^(3/2) =5^(3/2) =(√(125)) =5(√5)
$$\mathrm{2}^{\mathrm{3p}} =\left(\mathrm{2}^{\mathrm{2}} \right)^{\frac{\mathrm{3p}}{\mathrm{2}}} =\left(\mathrm{4}^{\mathrm{p}} \right)^{\mathrm{3}/\mathrm{2}} =\mathrm{5}^{\mathrm{3}/\mathrm{2}} =\sqrt{\mathrm{125}}\:=\mathrm{5}\sqrt{\mathrm{5}} \\ $$
Answered by BaliramKumar last updated on 16/Apr/24
2^(2p) = 5 ⇒ 2 = 5^(1/(2p)) 2^(3p) = (5^(1/(2p)) )^(3p) = 5^((3p)/(2p)) = 5^(3/2) = 5(√5)
$$\mathrm{2}^{\mathrm{2p}} \:=\:\mathrm{5}\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:\mathrm{2}\:=\:\mathrm{5}^{\frac{\mathrm{1}}{\mathrm{2p}}} \\ $$$$\mathrm{2}^{\mathrm{3p}} \:=\:\left(\mathrm{5}^{\frac{\mathrm{1}}{\mathrm{2p}}} \right)^{\mathrm{3p}} \:=\:\mathrm{5}^{\frac{\mathrm{3p}}{\mathrm{2p}}} \:=\:\mathrm{5}^{\frac{\mathrm{3}}{\mathrm{2}}} \:=\:\mathrm{5}\sqrt{\mathrm{5}} \\ $$
Answered by Skabetix last updated on 16/Apr/24
(2^2 )^p =5 ⇔ 2^(2p) =5 ⇔(2^(2p) )^(3/2) =5^(3/2) ⇔2^(2p×(3/2)) =5^(3/2) ⇔2^(3p) =5^(3/2) =5(√5)
$$\left(\mathrm{2}^{\mathrm{2}} \right)^{{p}} =\mathrm{5} \\ $$$$\Leftrightarrow\:\mathrm{2}^{\mathrm{2}{p}} =\mathrm{5} \\ $$$$\Leftrightarrow\left(\mathrm{2}^{\mathrm{2}{p}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} =\mathrm{5}^{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$$$\Leftrightarrow\mathrm{2}^{\mathrm{2}{p}×\frac{\mathrm{3}}{\mathrm{2}}} =\mathrm{5}^{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$$$\Leftrightarrow\mathrm{2}^{\mathrm{3}{p}} =\mathrm{5}^{\frac{\mathrm{3}}{\mathrm{2}}} =\mathrm{5}\sqrt{\mathrm{5}} \\ $$

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *