Find the geometric mean between :
`2a` and `8a^(3)`

Answer

To find the geometric mean between the terms \(2a\) and \(8a^3\), we can follow these steps:

### Step 1: Write the formula for the geometric mean
The geometric mean \(G\) of two numbers \(x_1\) and \(x_2\) is given by the formula:
\[ G = \sqrt{x_1 \cdot x_2} \]

### Step 2: Identify the terms
Here, our terms are:
- \(x_1 = 2a\)
- \(x_2 = 8a^3\)

### Step 3: Substitute the terms into the formula
Now we substitute \(x_1\) and \(x_2\) into the formula:
\[ G = \sqrt{(2a) \cdot (8a^3)} \]

### Step 4: Multiply the terms inside the square root
Next, we multiply the terms:
\[ G = \sqrt{2a \cdot 8a^3} = \sqrt{(2 \cdot 8)(a \cdot a^3)} \]
Calculating the multiplication:
\[ 2 \cdot 8 = 16 \quad \text{and} \quad a \cdot a^3 = a^{1+3} = a^4 \]
So we have:
\[ G = \sqrt{16a^4} \]

### Step 5: Simplify the square root
Now we simplify the square root:
\[ G = \sqrt{16} \cdot \sqrt{a^4} = 4 \cdot a^2 \]
Thus, the geometric mean is:
\[ G = 4a^2 \]

### Final Answer
The geometric mean between \(2a\) and \(8a^3\) is \(4a^2\).

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