Question

(3/5)^x(5/3)^2x= (5/3)^3

Answer 1

Answer:

$\left(\dfrac{3}{5}\right)^x\left(\dfrac{5}{3}\right)^{2x}=\left(\dfrac{5}{3}\right)^3\quad :\quad x=3$

Step-by-step explanation:

$\left(\dfrac{3}{5}\right)^x\left(\dfrac{5}{3}\right)^{2x}=\left(\dfrac{5}{3}\right)^3$

$\gray{\mathrm{Convert\:}\left(\dfrac{3}{5}\right)^x\mathrm{\:to\:base\:}\left(\dfrac{5}{3}\right)}$

$\gray{\left(\dfrac{3}{5}\right)^x=\left(\left(\dfrac{5}{3}\right)^{-1}\right)^x}$

$\left(\left(\dfrac{5}{3}\right)^{-1}\right)^x\left(\dfrac{5}{3}\right)^{2x}=\left(\dfrac{5}{3}\right)^3$

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

$\gray{\left(\left(\dfrac{5}{3}\right)^{-1}\right)^x=\left(\dfrac{5}{3}\right)^{-1x}}$

$\left(\dfrac{5}{3}\right)^{-1\cdot \:x}\left(\dfrac{5}{3}\right)^{2x}=\left(\dfrac{5}{3}\right)^3$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \:a^b\cdot \:a^c=a^{b+c}}$

$\gray{\left(\dfrac{5}{3}\right)^{-1x}\left(\dfrac{5}{3}\right)^{2x}=\left(\dfrac{5}{3}\right)^{-1x+2x}}$

$\left(\dfrac{5}{3}\right)^{-1\cdot \:x+2x}=\left(\dfrac{5}{3}\right)^3$

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

$-1\cdot \:x+2x=3$

$\gray{\mathrm{Simplify}}$

$x=3$