Simplify and write in exponential form :
`(16xx25xx5^(2)xxt^(8))/(10^(3)xxt^(4))`

Options

A. `10t^(2)`
B. `10t^(3)`
C. `10t^(4)`
D. `10t^(5)`

Answer

To simplify the expression \((16 \times 25 \times 5^{2} \times t^{8}) / (10^{3} \times t^{4})\) and write it in exponential form, we can follow these steps:

### Step 1: Break down the numbers into their prime factors
- \(16 = 2^4\) (since \(16 = 2 \times 2 \times 2 \times 2\))
- \(25 = 5^2\) (since \(25 = 5 \times 5\))
- \(10 = 2 \times 5\), so \(10^3 = (2 \times 5)^3 = 2^3 \times 5^3\)

### Step 2: Rewrite the expression using the prime factors
Now we can rewrite the original expression:
\[ \frac{(2^4 \times 5^2 \times 5^2 \times t^8)}{(2^3 \times 5^3 \times t^4)} \]

### Step 3: Combine the terms in the numerator
Combine the \(5\) terms in the numerator:
\[ 5^2 \times 5^2 = 5^{2 + 2} = 5^4 \]
Now the expression looks like:
\[ \frac{(2^4 \times 5^4 \times t^8)}{(2^3 \times 5^3 \times t^4)} \]

### Step 4: Simplify the expression
Now we can simplify the expression by dividing the powers of the same base:
- For \(2\):
\[ \frac{2^4}{2^3} = 2^{4-3} = 2^1 = 2 \]
- For \(5\):
\[ \frac{5^4}{5^3} = 5^{4-3} = 5^1 = 5 \]
- For \(t\):
\[ \frac{t^8}{t^4} = t^{8-4} = t^4 \]

### Step 5: Combine the simplified terms
Now we can combine all the simplified terms:
\[ 2 \times 5 \times t^4 \]

### Step 6: Write in exponential form
Since \(2 \times 5 = 10\), we can write the final expression as:
\[ 10 \times t^4 \]

### Final Answer
In exponential form, the simplified expression is:
\[ 10^1 \times t^4 \]