Let's analyze your friend's solution step by step:
The original equation is:
\[ 4^{(5x - 3)} = 16^{(x + 6)} \]
First, we can express \(16\) as \(4^2\):
\[ 16^{(x + 6)} = (4^2)^{(x + 6)} \]
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we can rewrite it:
\[ (4^2)^{(x + 6)} = 4^{2(x + 6)} \]
Now we can rewrite the equation:
\[ 4^{(5x - 3)} = 4^{2(x + 6)} \]
Since the bases are the same (both sides are powers of 4), we can set the exponents equal to each other:
\[ 5x - 3 = 2(x + 6) \]
Now, let's distribute on the right side:
\[ 5x - 3 = 2x + 12 \]
Next, we isolate \(x\). Subtract \(2x\) from both sides:
\[ 5x - 2x - 3 = 12 \]
This simplifies to:
\[ 3x - 3 = 12 \]
Now add 3 to both sides:
\[ 3x = 15 \]
Finally, divide by 3:
\[ x = 5 \]
So your friend made an error when simplifying the equation. The correct solution is:
\[ x = 5 \]
In conclusion, your friend's answer of \(x = \frac{9}{4}\) is incorrect.