The order of `(2)^(30)` is approximately :

Options

A. `10^(9)`
B. `10^(10)`
C. `10^(15)`
D. `10^(20)`

Answer

To find the order of \(2^{30}\), we can follow these steps:

### Step 1: Understand the concept of order of magnitude
The order of magnitude of a number is a way of expressing how large or small a number is, typically in powers of 10. For example, \(10^3\) is the order of magnitude for 1000.

### Step 2: Recall the value of \(2^{10}\)
It is known that:
\[ 2^{10} \approx 1024 \]
This is approximately equal to \(10^3\) (since \(1024\) is close to \(1000\)).

### Step 3: Express \(2^{30}\) in terms of \(2^{10}\)
We can express \(2^{30}\) as:
\[ 2^{30} = (2^{10})^3 \]

### Step 4: Substitute the approximate value of \(2^{10}\)
Since \(2^{10} \approx 10^3\), we can substitute this into our equation:
\[ 2^{30} \approx (10^3)^3 \]

### Step 5: Simplify the expression
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ (10^3)^3 = 10^{3 \cdot 3} = 10^9 \]

### Step 6: Conclusion
Thus, the order of \(2^{30}\) is approximately:
\[ 10^9 \]

### Final Answer
The order of \(2^{30}\) is approximately \(10^9\).

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