Question

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

Answer 1

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

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 2

{Solution}

FORMULA TO BE IMPLEMENTED

1. \: \: \: \sf{ {a}^{m} \times {a}^{n} = {a}^{m + n} \: }1.am×an=am+n

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

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} }(12564)2×(54)4×(2516)2x+1=(625256)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} }⟹[(54)3]2×(54)4×[(54)2]2x+1=[(54)4]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} }⟹(54)6×(54)4×(54)4x+2=(54)12x

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

\implies \: \displaystyle \sf{ 4x +1 2 = 12x}⟹4x+12=12x

\implies \: \displaystyle \sf{8x = 12}⟹8x=12

\implies \: \displaystyle \sf{ x = \frac{3}{2} }⟹x=23

RESULT

The answer is

\boxed{ \: \displaystyle \sf{ \: x = \frac{3}{2} \: \: \: }}x=23