`3sqrt(2^(5))sqrt(4^(9))sqrt(8)=`

Options

A. 1.9
B. `2.0`
C. `2.1`
D. `2.3`

Explanation

To solve the expression \( 3\sqrt{2^5}\sqrt{4^9}\sqrt{8} \), we can break it down step by step.

### Step 1: Rewrite the square roots in exponential form
We can express the square roots in terms of exponents:
\[ 3\sqrt{2^5} = 3 \cdot (2^5)^{1/2} = 3 \cdot 2^{5/2} \]
\[ \sqrt{4^9} = (4^9)^{1/2} = 4^{9/2} \]
\[ \sqrt{8} = 8^{1/2} \]

### Step 2: Rewrite \(4\) and \(8\) in terms of base \(2\)
Next, we rewrite \(4\) and \(8\) as powers of \(2\):
\[ 4 = 2^2 \quad \text{so} \quad 4^{9/2} = (2^2)^{9/2} = 2^{2 \cdot \frac{9}{2}} = 2^9 \]
\[ 8 = 2^3 \quad \text{so} \quad 8^{1/2} = (2^3)^{1/2} = 2^{3/2} \]

### Step 3: Combine all terms
Now we can combine all the terms:
\[ 3\sqrt{2^5}\sqrt{4^9}\sqrt{8} = 3 \cdot 2^{5/2} \cdot 2^9 \cdot 2^{3/2} \]

### Step 4: Use the properties of exponents
Using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\), we can combine the powers of \(2\):
\[ 2^{5/2} \cdot 2^9 \cdot 2^{3/2} = 2^{\left(\frac{5}{2} + 9 + \frac{3}{2}\right)} \]

### Step 5: Simplify the exponent
Now we need to simplify the exponent:
\[ \frac{5}{2} + 9 + \frac{3}{2} = \frac{5}{2} + \frac{18}{2} + \frac{3}{2} = \frac{5 + 18 + 3}{2} = \frac{26}{2} = 13 \]
So we have:
\[ 3 \cdot 2^{13} \]

### Step 6: Calculate the final result
Now we can calculate \(3 \cdot 2^{13}\):
\[ 2^{13} = 8192 \quad \text{so} \quad 3 \cdot 8192 = 24576 \]

Thus, the final answer is:
\[ \boxed{24576} \]