To solve the equation [tex]\( 36^{5x + 4} = 6^{x^2 + 10x - 56} \)[/tex], we will start by simplifying the bases. Notice that:
[tex]\[ 36 = 6^2 \][/tex]
Thus, we can rewrite [tex]\( 36^{5x + 4} \)[/tex] as:
[tex]\[ (6^2)^{5x + 4} = 6^{2(5x + 4)} = 6^{10x + 8} \][/tex]
So, our original equation becomes:
[tex]\[ 6^{10x + 8} = 6^{x^2 + 10x - 56} \][/tex]
Because the bases are the same (both bases are 6), we can set the exponents equal to each other:
[tex]\[ 10x + 8 = x^2 + 10x - 56 \][/tex]
Next, we simplify the equation by moving all terms to one side:
[tex]\[ 0 = x^2 + 10x - 56 - 10x - 8 \][/tex]
This simplifies to:
[tex]\[ 0 = x^2 - 64 \][/tex]
This is now a simple quadratic equation. We solve it by factoring:
[tex]\[ x^2 - 64 = (x - 8)(x + 8) = 0 \][/tex]
Setting each factor equal to zero gives us:
[tex]\[ x - 8 = 0 \quad \text{or} \quad x + 8 = 0 \][/tex]
Thus, the solutions are:
[tex]\[ x = 8 \quad \text{or} \quad x = -8 \][/tex]
Therefore, the solutions to the equation are:
[tex]\[ x = 8, -8 \][/tex]
