To simplify the expression ( 16^{m+4} + 8^{m+4} - 4^{m+4} ), let's first rewrite each term using powers of 2:
- ( 16^{m+4} = (2^4)^{m+4} = 2^{4(m+4)} = 2^{4m + 16} )
- ( 8^{m+4} = (2^3)^{m+4} = 2^{3(m+4)} = 2^{3m + 12} )
- ( 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)
]
