Find the value of the expression `(10^(-1)xx5^(x-3)xx4^(x-1))/(10xx5^(x-5)xx4^(x-2))`

Answer

To find the value of the expression

\[ \frac{10^{-1} \times 5^{x-3} \times 4^{x-1}}{10 \times 5^{x-5} \times 4^{x-2}}, \]

we will simplify it step by step.

### Step 1: Rewrite the expression using properties of exponents

We can use the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\). We will apply this property to each base (10, 5, and 4) in the expression.

\[ = \frac{10^{-1}}{10^1} \times \frac{5^{x-3}}{5^{x-5}} \times \frac{4^{x-1}}{4^{x-2}} \]

### Step 2: Simplify each part

1. For the base 10:
\[ 10^{-1} - 1 = 10^{-2} \]

2. For the base 5:
\[ (x-3) - (x-5) = -3 + 5 = 2 \quad \Rightarrow \quad 5^2 \]

3. For the base 4:
\[ (x-1) - (x-2) = -1 + 2 = 1 \quad \Rightarrow \quad 4^1 \]

Putting it all together, we have:

\[ = 10^{-2} \times 5^2 \times 4^1 \]

### Step 3: Substitute the values

Now substituting the values we calculated:

\[ = 10^{-2} \times 25 \times 4 \]

### Step 4: Calculate \(10^{-2}\)

Using the property \(a^{-m} = \frac{1}{a^m}\):

\[ 10^{-2} = \frac{1}{10^2} = \frac{1}{100} \]

### Step 5: Combine the values

Now, we can combine everything:

\[ = \frac{1}{100} \times 25 \times 4 \]

Calculating \(25 \times 4\):

\[ 25 \times 4 = 100 \]

So now we have:

\[ = \frac{100}{100} = 1 \]

### Final Answer

The value of the expression is

\[ \boxed{1}. \]