Question 414909
If the equation is
{{{27^(2x+1) = 9^(4x)}}}
then please enclose multiple term exponents in parentheses.<br>
Solving equations where the variable is in an exponent is usually done in one of two ways:<ul><li>Rewrite both sides as powers of the same number. This cannot be done all the time. But this way is faster/easier when it can be done.</li><li>Using logarithms.</li></ul>
Since 27 and 9 are both powers of 3, the first ways is available to us on this equation:
{{{(3^3)^(2x+1) = (3^2)^(4x)}}}
The rule for exponents when raising a power to a power is to multiply the exponents:
{{{3^(3*(2x+1)) = 3^(2*4x)}}}
which simplifies to:
{{{3^(6x+3) = 3^(8x)}}}
This equation now says that two powers of 3 are equal. The only way this can be true is if the exponents are equal, too. So:
6x+3 = 8x
This is very easy to solve. Subtracting 6x from each side:
3 = 2x
Dividing by 2 we get:
3/2 = x
This is the solution to your equation.