16^m+4+8^m+4-4^m+4

To simplify the expression \( 16^{m+4} + 8^{m+4} - 4^{m+4} \), let's first rewrite each term using powers of 2:

1. \( 16^{m+4} = (2^4)^{m+4} = 2^{4(m+4)} = 2^{4m + 16} \)
2. \( 8^{m+4} = (2^3)^{m+4} = 2^{3(m+4)} = 2^{3m + 12} \)
3. \( 4^{m+4} = (2^2)^{m+4} = 2^{2(m+4)} = 2^{2m + 8} \)

Now, substituting these back into the expression:

\[ 16^{m+4} + 8^{m+4} - 4^{m+4} = 2^{4m + 16} + 2^{3m + 12} - 2^{2m + 8} \]

Next, let's find a common exponent to factor the expression. Notice that the smallest exponent among \( 4m + 16 \), \( 3m + 12 \), and \( 2m + 8 \) is \( 2m + 8 \).

Now we can factor out \( 2^{2m + 8} \):

\[ 2^{2m + 8} \left( 2^{4m + 16 - (2m + 8)} + 2^{3m + 12 - (2m + 8)} - 1 \right) \]

Calculating the exponents in the first term:

\[ 4m + 16 - (2m + 8) = 2m + 8 \]

Then in the second term:

\[ 3m + 12 - (2m + 8) = m + 4 \]

Now we can rewrite the expression:

\[ 2^{2m + 8} \left( 2^{2m + 8} + 2^{m + 4} - 1 \right) \]

So, the simplified form of the expression \( 16^{m+4} + 8^{m+4} - 4^{m+4} \) is:

\[ 2^{2m + 8} \left( 2^{2m + 8} + 2^{m + 4} - 1 \right) \]