Question 1195385
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{{{root(3,(81^2 ))/root(2,(27^x ))=9}}}<br>
Rewrite using powers of 3:<br>
{{{root(3,((3^4)^2 ))/root(2,((3^3)^x ))=3^2}}}<br>
Use the power to a power rule: {{{(x^a)^b=x^(ab)}}}<br>
{{{root(3,(3^8 ))/root(2,(3^(3x) ))=3^2}}}<br>
Rewrite using fractional exponents:<br>
{{{(3^8)^(1/3)/(3^(3x))^(1/2)=3^2}}}<br>
Use the power to a power rule again:
{{{(3^(8/3))/(3^(3x/2))=3^2}}}<br>
Clear fractions:<br>
{{{(3^(8/3))=(3^2)(3^(3x/2))}}}<br>
Multiply on the right (same base, so add the exponents):<br>
{{{(3^(8/3))=3^(2+3x/2)}}}<br>
The bases are the same, so the exponents are the same:<br>
{{{8/3=2+3x/2}}}<br>
Clear fractions (multiply by the LCD, 6):<br>
{{{16=12+9x}}}
{{{9x=4}}}
{{{x=4/9}}}<br>
ANSWER: x = 4/9<br>
Note that the sequence of operations could be in many different orders; or possibly grouped to perform more than one operation at a time.<br>
I chose to take small steps in an arbitrary order....<br>