Simplify and write the in the exponential form:
`((-2)^(4)xx(2^(3))^(3))/((2^(4))^(2))`

Options

A. `2^(5)`
B. `2^(6)`
C. `2^(7)`
D. `2^(8)`

Answer

To simplify the expression \(\frac{(-2)^{4} \times (2^{3})^{3}}{(2^{4})^{2}}\) and write it in exponential form, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{(-2)^{4} \times (2^{3})^{3}}{(2^{4})^{2}} \]

### Step 2: Simplify the numerator
First, simplify \((2^{3})^{3}\) using the power of a power rule, which states that \((a^{m})^{n} = a^{m \cdot n}\):
\[ (2^{3})^{3} = 2^{3 \cdot 3} = 2^{9} \]
Now, substitute this back into the expression:
\[ \frac{(-2)^{4} \times 2^{9}}{(2^{4})^{2}} \]

### Step 3: Simplify the denominator
Now simplify \((2^{4})^{2}\) using the same power of a power rule:
\[ (2^{4})^{2} = 2^{4 \cdot 2} = 2^{8} \]
Now substitute this back into the expression:
\[ \frac{(-2)^{4} \times 2^{9}}{2^{8}} \]

### Step 4: Simplify \((-2)^{4}\)
Since \((-2)^{4}\) is a negative number raised to an even power, it becomes positive:
\[ (-2)^{4} = 2^{4} \]
Now substitute this back into the expression:
\[ \frac{2^{4} \times 2^{9}}{2^{8}} \]

### Step 5: Combine the terms in the numerator
Using the property of exponents that states \(a^{m} \times a^{n} = a^{m+n}\):
\[ 2^{4} \times 2^{9} = 2^{4 + 9} = 2^{13} \]
Now the expression becomes:
\[ \frac{2^{13}}{2^{8}} \]

### Step 6: Apply the quotient rule
Using the property of exponents that states \(\frac{a^{m}}{a^{n}} = a^{m-n}\):
\[ \frac{2^{13}}{2^{8}} = 2^{13 - 8} = 2^{5} \]

### Final Answer
Thus, the simplified expression in exponential form is:
\[ 2^{5} \]