Question

D%7B9%7D%20%29%5E%7B5%7D%20%20%3D%20%28%20%5Cfrac%7B4%7D%7B3%7D%20%29%5E%7Bx%20%2B%202%7D%20.%20%5C%3A%20find%20%5C%3A%20x" id="TexFormula1" title="if \: ( \frac{3}{4} )^{6} \times ( \frac{16}{9} )^{5} = ( \frac{4}{3} )^{x + 2} . \: find \: x" alt="if \: ( \frac{3}{4} )^{6} \times ( \frac{16}{9} )^{5} = ( \frac{4}{3} )^{x + 2} . \: find \: x" align="absmiddle" class="latex-formula">
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Answer 1

Answer:

2

Step-by-step explanation:

$(\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{x+2}$

$\frac{3^6}{4^6} \cdot \frac{16^5}{9^5}=\frac{4^{x+2}}{3^{x+2}}$

$\frac{3^6}{4^6} \cdot \frac{(4^2)^5}{(3^2)^5}=\frac{4^{x+2}}{3^{x+2}}$

$\frac{3^6}{4^6} \cdot \frac{4^{10}}{3^{10}}=\frac{4^{x+2}}{3^{x+2}}$

$\frac{3^6}{3^{10}} \cdot \frac{4^{10}}{4^6}=\frac{4^{x+2}}{3^{x+2}}$

$3^{-4} \cdot 4^{4}=4^{x+2}3^{-(x+2)}$

This implies

x+2=4

and

-(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=4

It is the same equation as the other.

You will get x=2 either way.

Let's check:

$(\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{2+2}$

$(\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{4}$

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.