Question

Determine (8x)^x if 9^x+2=240+9^x

Answer 1

GIVEN:-

• $\rm{9^{x+2} = 240 + 9x}$

TO FIND:-

• Determine$\rm{ (8x)^x}$

SOLUTION:-

$\implies\tt{9^{x+2} = 240 + 9^x}$

$\implies\tt{ 9^x \times{9^2} = 240 + 9^x}$

$\implies\tt{9^x\times{81} = 240 + 9^x}$

$\implies\tt{ 9^x\times{81} - 9^x = 240}$

$\implies\tt{ 9^x(81-1)= 240 }$

$\implies\tt{ 9^x\times{80} = 240}$

$\implies\tt{ 9^x = \dfrac{240}{80}}$

$\implies\tt{ 9^x = 3}$

$\implies\tt{ 3^{2x} = 3}$

• Bases are same , equating the powers

$\implies\tt{ 2x = 1}$

$\implies\tt{ x = \dfrac{1}{2}}$.

Therefore,

$\implies\tt{ (8x)^x = (8\times{\dfrac{1}{2}})^2}$

$\implies\tt{ (8x)^x = (4)^{\frac{1}{2}}}$

$\implies\tt{ (8x)^x = 2^{2\times{\frac{1}{2}}}}$

$\implies\tt{(8x)^x = 2}$

Answer 2

ᎯᏁᏕᏯᎬᏒ

${\star} ~\red {\sf {{9}^{x + 2} = 240 + {9}^{x} }}$

$⇢\sf {9}^{x} \times {9}^{2} = 240 + {9}^{x}$

$⇢ \sf {9}^{x} \times 81 = 240 + {9}^{x}$

$⇢ \sf {9}^{x} \times 81 - {9}^{x} = 240$

$⇢\sf {9}^{x} (81 - 1) = 240$

$⇢ \sf {9}^{x} \times 80 = 240$

$⇢ \sf {9}^{x} = \frac{240}{80}$

$⇢ \sf {9}^{x} = 3$

$⇢ \sf {({3}^{2} ) }^{x} = 3$

$⇢\sf {3}^{2x} = {3}^{1}$$⇢ \sf 2x = 1$

$⇢ \sf x = \frac{1}{2}$

$\purple{.........................................................}$

$\sf {8x}^{x} = {(8 \times \frac{1}{2}) }^{ \frac{1}{2} }$

$⇢ \sf {8x}^{x} = {4}^{ \frac{1}{2} }$

$⇢ \sf {8x}^{x} = \sqrt{4}$

${\therefore}~ \red{ \sf {{8x}^{x} = 2}}$