Simplify : `(25 xx a^9)/( 5^(-3) xx 10xx a^(-18) )`

Options

A. `(625a^(27) )/( 12)`
B. `(625 a^5)/( 20)`
C. `625 a`
D. `(625a^(27) )/(2)`

Answer

To simplify the expression \((25 \times a^9)/(5^{-3} \times 10 \times a^{-18})\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{25 \times a^9}{5^{-3} \times 10 \times a^{-18}} \]

### Step 2: Break down the denominator
We know that \(10\) can be expressed as \(2 \times 5\). So, we rewrite the denominator:
\[ 5^{-3} \times 10 = 5^{-3} \times (2 \times 5) = 5^{-3} \times 2 \times 5^1 = 2 \times 5^{-2} \]

### Step 3: Substitute back into the expression
Now, substituting this back into the expression gives:
\[ \frac{25 \times a^9}{2 \times 5^{-2} \times a^{-18}} \]

### Step 4: Simplify the constants
Next, we can simplify the constants:
\[ 25 = 5^2 \quad \text{(since \(25 = 5 \times 5\))} \]
So the expression becomes:
\[ \frac{5^2 \times a^9}{2 \times 5^{-2} \times a^{-18}} \]

### Step 5: Simplify the powers of 5
Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify \(5^2\) and \(5^{-2}\):
\[ \frac{5^2}{5^{-2}} = 5^{2 - (-2)} = 5^{2 + 2} = 5^4 \]

### Step 6: Simplify the powers of \(a\)
Now, we simplify the powers of \(a\):
\[ \frac{a^9}{a^{-18}} = a^{9 - (-18)} = a^{9 + 18} = a^{27} \]

### Step 7: Combine everything
Putting it all together, we have:
\[ \frac{5^4 \times a^{27}}{2} \]

### Step 8: Calculate \(5^4\)
Calculating \(5^4\):
\[ 5^4 = 625 \]

### Final Result
Thus, the simplified expression is:
\[ \frac{625 \times a^{27}}{2} = \frac{625a^{27}}{2} \]

### Summary
The final simplified form of the given expression is:
\[ \frac{625a^{27}}{2} \]