Simplify and express each of the following in exponential form:
`(2^(8)xxa^5)/(4^(3)xxa^3)`

Answer

To simplify the expression \((2^{8} \cdot a^{5})/(4^{3} \cdot a^{3})\) and express it in exponential form, follow these steps:

### Step 1: Rewrite the base 4 in terms of base 2
We know that \(4\) can be expressed as \(2^{2}\). Therefore, we can rewrite \(4^{3}\) as:
\[ 4^{3} = (2^{2})^{3} \]
Using the property of exponents \((p^{m})^{n} = p^{m \cdot n}\), we can simplify this to:
\[ (2^{2})^{3} = 2^{2 \cdot 3} = 2^{6} \]

### Step 2: Substitute back into the expression
Now, substitute \(4^{3}\) with \(2^{6}\) in the original expression:
\[ \frac{2^{8} \cdot a^{5}}{4^{3} \cdot a^{3}} = \frac{2^{8} \cdot a^{5}}{2^{6} \cdot a^{3}} \]

### Step 3: Apply the property of exponents for division
Using the property of exponents \(\frac{p^{m}}{p^{n}} = p^{m - n}\), we can simplify the expression:
\[ \frac{2^{8}}{2^{6}} = 2^{8 - 6} = 2^{2} \]
And for the \(a\) terms:
\[ \frac{a^{5}}{a^{3}} = a^{5 - 3} = a^{2} \]

### Step 4: Combine the results
Now, combine the results from the previous steps:
\[ \frac{2^{8} \cdot a^{5}}{2^{6} \cdot a^{3}} = 2^{2} \cdot a^{2} \]

### Final Expression
Thus, the simplified expression in exponential form is:
\[ 2^{2} \cdot a^{2} \]