for what value of x does $64^{3x}=512^{2x + 12}$?\n1\n3\n12\nno solution
for what value of x does $64^{3x}=512^{2x + 12}$?\n1\n3\n12\nno solution
Answer
Explanation:
Step1: Rewrite bases as powers of 2
Since $64 = 2^6$ and $512=2^9$, the equation $64^{3x}=512^{2x + 12}$ can be rewritten as $(2^6)^{3x}=(2^9)^{2x + 12}$.
Step2: Apply power - of - a - power rule
According to the power - of - a - power rule $(a^m)^n=a^{mn}$, we have $2^{6\times3x}=2^{9\times(2x + 12)}$, which simplifies to $2^{18x}=2^{18x+108}$.
Step3: Set exponents equal
If $a^m=a^n$, then $m = n$. So, $18x=18x + 108$.
Step4: Solve for x
Subtract $18x$ from both sides: $18x-18x=18x + 108-18x$, resulting in $0 = 108$, which is a contradiction.
Answer:
no solution