Simplify (64/125)^2×(4/5)^4×(16/25)^2x+1=(256/625)^3x

1.	Simplify (64/125)^2×(4/5)^4×(16/25)^2x+1=(256/625)^3x
Answer» $\displaystyle\huge\red{\underline{\underline{Solution}}}$ FORMULA TO BE IMPLEMENTED $1. \: \: \: \sf{ {a}^{m} \times {a}^{n} = {a}^{m + n} \: }$ $2. \: \: \sf{{ ({a}^{m})}^{n} = {a}^{m n} \: }$ TO DETERMINE The VALUE of X when $\displaystyle \sf{ { \bigg( \frac{64}{125} \bigg)}^{2} \times { \bigg( \frac{4}{5} \bigg)}^{4} \times { \bigg( \frac{16}{25} \bigg)}^{2x + 1}\: ={ \bigg( \frac{256}{625} \bigg)}^{3x} }$ CALCULATION $\displaystyle \implies \: \sf{ { \bigg[ {\bigg( \frac{4}{5} \bigg)}^{3} \bigg]}^{2} \times { \bigg( \frac{4}{5} \bigg)}^{4} \times { { \bigg[ {\bigg( \frac{4}{5} \bigg)}^{2} \bigg]}}^{2x + 1}\: ={ { \bigg[ {\bigg( \frac{4}{5} \bigg)}^{4} \bigg]}}^{3x} }$ $\implies \: \displaystyle \sf{ { \bigg( \frac{4}{5} \bigg)}^{6} \times { \bigg( \frac{4}{5} \bigg)}^{4} \times { \bigg( \frac{4}{5} \bigg)}^{4x + 2}\: ={ \bigg( \frac{4}{5} \bigg)}^{12x} }$ $\implies \: \displaystyle \sf{ { \bigg( \frac{4}{5} \bigg)}^{4x + 12} ={ \bigg( \frac{4}{5} \bigg)}^{12x} }$ $\implies \: \displaystyle \sf{ 4x +1 2 = 12x}$ $\implies \: \displaystyle \sf{8x = 12}$ $\implies \: \displaystyle \sf{ x = \frac{3}{2} }$ RESULT The ANSWER is $\boxed{ \: \displaystyle \sf{ \: x = \frac{3}{2} \: \: \: }}$

Answer»

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

$1. \: \: \: \sf{ {a}^{m} \times {a}^{n} = {a}^{m + n} \: }$

$2. \: \: \sf{{ ({a}^{m})}^{n} = {a}^{m n} \: }$

TO DETERMINE

The VALUE of X when

$\displaystyle \sf{ { \bigg( \frac{64}{125} \bigg)}^{2} \times { \bigg( \frac{4}{5} \bigg)}^{4} \times { \bigg( \frac{16}{25} \bigg)}^{2x + 1}\: ={ \bigg( \frac{256}{625} \bigg)}^{3x} }$

CALCULATION

$\displaystyle \implies \: \sf{ { \bigg[ {\bigg( \frac{4}{5} \bigg)}^{3} \bigg]}^{2} \times { \bigg( \frac{4}{5} \bigg)}^{4} \times { { \bigg[ {\bigg( \frac{4}{5} \bigg)}^{2} \bigg]}}^{2x + 1}\: ={ { \bigg[ {\bigg( \frac{4}{5} \bigg)}^{4} \bigg]}}^{3x} }$

$\implies \: \displaystyle \sf{ { \bigg( \frac{4}{5} \bigg)}^{6} \times { \bigg( \frac{4}{5} \bigg)}^{4} \times { \bigg( \frac{4}{5} \bigg)}^{4x + 2}\: ={ \bigg( \frac{4}{5} \bigg)}^{12x} }$