[tex]if \: ( \frac{3}{4} )^{6} \times ( \frac{16}{9} )^{5} = ( \frac{4}{3} )^{x + 2} . \: find \: x[/tex]​

Accepted Solution

Answer:2Step-by-step explanation:[tex](\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{x+2}[/tex][tex]\frac{3^6}{4^6} \cdot \frac{16^5}{9^5}=\frac{4^{x+2}}{3^{x+2}}[/tex][tex]\frac{3^6}{4^6} \cdot \frac{(4^2)^5}{(3^2)^5}=\frac{4^{x+2}}{3^{x+2}}[/tex][tex]\frac{3^6}{4^6} \cdot \frac{4^{10}}{3^{10}}=\frac{4^{x+2}}{3^{x+2}}[/tex][tex]\frac{3^6}{3^{10}} \cdot \frac{4^{10}}{4^6}=\frac{4^{x+2}}{3^{x+2}}[/tex][tex]3^{-4} \cdot 4^{4}=4^{x+2}3^{-(x+2)}[/tex]This implies x+2=4and-(x+2)=-4.x+2=4 implies x=2 since subtract 2 on both sides gives us x=2.Solving -(x+2)=-4 should give us the same value.Multiply both sides by -1:x+2=4It is the same equation as the other.You will get x=2 either way.Let's check:[tex](\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{2+2}[/tex][tex](\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{4}[/tex]Put both sides into your calculator and see if you get the same thing on both sides:Left hand side gives 256/81. Right hand side gives 256/81.Both side are indeed the same for x=2.