Simplify `(8^(3a)xx2^(5)xx2^(2a))/(4xx2^(11a)xx2^(-2a))`

Answer

To simplify the expression \((8^{3a} \times 2^{5} \times 2^{2a}) / (4 \times 2^{11a} \times 2^{-2a})\), we will follow these steps:

### Step 1: Rewrite the bases in terms of powers of 2
We know that:
- \(8 = 2^3\)
- \(4 = 2^2\)

Thus, we can rewrite the expression as:
\[ \frac{(2^3)^{3a} \times 2^{5} \times 2^{2a}}{2^{2} \times 2^{11a} \times 2^{-2a}} \]

### Step 2: Apply the power of a power rule
Using the identity \((a^b)^c = a^{bc}\), we can simplify \((2^3)^{3a}\):
\[ (2^3)^{3a} = 2^{3 \cdot 3a} = 2^{9a} \]

Now the expression becomes:
\[ \frac{2^{9a} \times 2^{5} \times 2^{2a}}{2^{2} \times 2^{11a} \times 2^{-2a}} \]

### Step 3: Combine the powers in the numerator
Using the identity \(a^b \times a^c = a^{b+c}\), we combine the powers in the numerator:
\[ 2^{9a + 5 + 2a} = 2^{11a + 5} \]

### Step 4: Combine the powers in the denominator
Similarly, we combine the powers in the denominator:
\[ 2^{2 + 11a - 2a} = 2^{11a - 2a + 2} = 2^{9a + 2} \]

### Step 5: Apply the quotient rule for exponents
Using the identity \(\frac{a^b}{a^c} = a^{b-c}\), we can simplify the expression:
\[ \frac{2^{11a + 5}}{2^{9a + 2}} = 2^{(11a + 5) - (9a + 2)} \]

### Step 6: Simplify the exponent
Now we simplify the exponent:
\[ (11a + 5) - (9a + 2) = 11a - 9a + 5 - 2 = 2a + 3 \]

### Final Result
Thus, the simplified expression is:
\[ 2^{2a + 3} \]

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