Certainly! Let's solve the equation step-by-step.
The given equation is:
[tex]\[ 36^{5x + 4} = 6^{x^2 + 10x - 56} \][/tex]
To simplify things, let's express both sides with the same base. Note that:
[tex]\[ 36 = 6^2 \][/tex]
Thus, we can rewrite 36 as [tex]\(6^2\)[/tex], so the equation becomes:
[tex]\[ (6^2)^{5x + 4} = 6^{x^2 + 10x - 56} \][/tex]
Now apply the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex] to the left side:
[tex]\[ 6^{2(5x + 4)} = 6^{x^2 + 10x - 56} \][/tex]
Simplify the exponent on the left side:
[tex]\[ 6^{10x + 8} = 6^{x^2 + 10x - 56} \][/tex]
Since the bases on both sides are the same, we can equate the exponents:
[tex]\[ 10x + 8 = x^2 + 10x - 56 \][/tex]
Now, let's solve for [tex]\(x\)[/tex]. First, subtract [tex]\(10x\)[/tex] from both sides:
[tex]\[ 8 = x^2 - 56 \][/tex]
Next, add 56 to both sides to isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ x^2 = 64 \][/tex]
To solve for [tex]\(x\)[/tex], take the square root of both sides:
[tex]\[ x = \pm \sqrt{64} \][/tex]
This gives us:
[tex]\[ x = \pm 8 \][/tex]
So, we have two solutions:
[tex]\[ x = 8 \quad \text{or} \quad x = -8 \][/tex]
Hence, the solutions are:
[tex]\[ x = 8, -8 \][/tex]
