Question 88347
Given:
.
{{{(7^(4n-5))*(7^(2n+3))^3=(7^(4-6n))^4}}}
.
First, note that all the bases are 7, so the rules that depend on the bases being common can
be used.
.
Next note that two of the terms have exponents. The applicable rule for this this is:
.
{{{(x^a)^b = x^(a*b)}}}
.
for the term {{{(7^(2n + 3))^3}}} you have x = 7, a = 2n +3, and b = 3.  Substituting
these values into the rule results in:
.
{{{(7^(2n+3))^3 = 7^((2n+3)*3) = 7^(6n + 9)}}}
.
Similarly, the other term of the original problem that also can be simplified using the
power rule is:
.
{{{(7^(4-6n))^4}}}
.
For this term x = 7, a = (4 - 6n), and b = 3. Substituting these values into the power rule
results in:
.
{{{(7^(4-6n))^4 = 7^((4-6n)*4) = 7^(16-24n)}}}
.
Substituting the two simplifications into the original given problem results in it becoming:
.
{{{7^(4n-5)*7^(6n + 9)=7^(16-24n)}}}
.
Now you can apply the rule for two terms of a common base, each term raised to an exponent, 
when the two terms are multiplied.  This rule states that the product is the common base
raised to the sum of the exponents. In equation form this is:
.
{{{x^a * x^b = x^(a+b)}}}
.
On the left side of the equation we have developed there are two terms that are multiplied.
In this product x = 7, a = 4n - 5, and b = 6n + 9. When you substitute these values into
the left side of the equation you get:
.
{{{7^(4n-5)*7^(6n + 9) = 7^((4n-5)+(6n + 9)) = 7^(10n + 4)}}}
.
This becomes the left side of the equation. The equation becomes:
.
{{{7^(10n + 4)= 7^(16-24n)}}}
.
Note that since the bases on the left and right sides are both the same ... 7 ... for
the equation to be true, the two exponents have to be the same. So therefore you can say:
.
{{{10n + 4 = 16 - 24n}}}
.
We can solve this equation by getting all the terms containing n on one side of the equation
and all the terms that are just numbers on the other side.  Start by adding +24n to both sides.
this makes the equation become:
.
{{{24n + 10n + 4 = 16 - 24n + 24n}}}
.
combining the terms containing n reduces the equation to:
.
{{{34n + 4 = 16}}}
.
Next add -4 to both sides to eliminate the +4 on the left side. This makes the equation:
.
{{{34n = 12}}}
.
Finally, solve for n by dividing both sides by 34 to get:
.
{{{n = 12/34 = 6/17}}}
.
Hope this helps you to see your way through the problem.