If \(4^{2x+1}=8^{x+3}\), then \(x=?\)

Given: \(4^{2x+1}=8^{x+3}\)

Step 1: Use the concept of prime factorization to write the number as the product of its power:

\(4=2×2=2^2\) and \(8=2×2×2=2^3\)

Therefore, \(4^{2x+1}=8^{x+3}\)

\(\Rightarrow (2^2)^{2x+1}=(2^3)^{x+3}\)

Step 2: Use the appropriate laws of exponents to simplify the expressions in in L.H.S and R.H.S:

\(\Rightarrow (2)^{2(2x+1)}=(2)^{3(x+3})\).  \( [ (a^m)^n=a^{mn} ]\)

\(\Rightarrow 2(2x+1)=3(x+3) \)

Step 3: Solve the equation for  x:

\(\Rightarrow 4x+2=3x+9 \)

\(\Rightarrow 4x-3x=9-2\)

\(\Rightarrow x=7\)

Therefore, \(x=7\).