Question

2^2^x=16^2^3x find the value of x

Answer 1

Answer:

$2^{2^x}=16^{2^{3x}}\quad :\quad x=-1$

Step-by-step explanation:

$2^{2^x}=16^{2^{3x}}$

$\gray{\mathrm{Convert\:}16^{2^{3x}}\mathrm{\:to\:base\:}2}$

$\gray{16^{2^{3x}}=\left(2^4\right)^{2^{3x}}}$

$\gray{2^{2^x}=\left(2^4\right)^{2^{3x}}}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}}$

$\gray{\left(2^4\right)^{2^{3x}}=2^{4\cdot \:2^{3x}}}$

$2^{2^x}=\left(2^4\right)^{2^{3x}}$

$\gray{\mathrm{If\:}a^{f\left(x\right)}=a^{g\left(x\right)}\mathrm{,\:then\:}f\left(x\right)=g\left(x\right)}$

$2^x=4\cdot \:2^{3x}$

$\mathrm{Simplify\:}4\cdot \:2^{3x}:\quad 2^{2+3x}$

$2^x=2^{2+3x}$

$\gray{\mathrm{If\:}a^{f\left(x\right)}=a^{g\left(x\right)}\mathrm{,\:then\:}f\left(x\right)=g\left(x\right)}$

$x=2+3x$

$\mathrm{Solve\:}\:x=2+3x$

$x=2+3x$

$\gray{\mathrm{Subtract\:}3x\mathrm{\:from\:both\:sides}}$

$x-3x=2+3x-3x$

$\gray{\mathrm{Simplify}}$

$-2x=2$

$\gray{\mathrm{Divide\:both\:sides\:by\:}-2}$

$\dfrac{-2x}{-2}=\dfrac{2}{-2}$

$\gray{\mathrm{Simplify}}$

$x=-1$