Simplify or otherwise reduce the following expression using the rules of exponents.
`2^(4)xx2^(5)div 2^(7)-2^(4)`

Answer

To simplify the expression \( 2^4 \times 2^5 \div 2^7 - 2^4 \), we will follow the rules of exponents step by step.

### Step 1: Apply the Division Rule
The first operation we need to perform is the division part of the expression. According to the laws of exponents, when we divide two exponential terms with the same base, we subtract the exponents.

\[ 2^4 \times 2^5 \div 2^7 = 2^{4 + 5 - 7} \]

### Step 2: Simplify the Exponents
Now, we simplify the exponent:

\[ 4 + 5 - 7 = 2 \]

Thus, we have:

\[ 2^{4 + 5 - 7} = 2^2 \]

### Step 3: Substitute Back into the Expression
Now we substitute \( 2^2 \) back into the expression:

\[ 2^2 - 2^4 \]

### Step 4: Factor Out Common Terms
Next, we notice that both terms have a common factor of \( 2^2 \):

\[ 2^2(1 - 2^{4 - 2}) = 2^2(1 - 2^2) \]

### Step 5: Simplify Further
Now we simplify \( 1 - 2^2 \):

\[ 2^2(1 - 4) = 2^2 \times (-3) \]

### Step 6: Calculate the Final Result
Finally, we calculate \( 2^2 \) and multiply it by \(-3\):

\[ 2^2 = 4 \quad \text{so} \quad 4 \times (-3) = -12 \]

Thus, the simplified expression is:

\[ \boxed{-12} \]

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