To simplify the expression [tex]\(\frac{2^{-3} \times 5^{-3} \times 10^2 \times 25}{5^4 \times 2^{-5}}\)[/tex], let's break it down into more manageable steps and work through the problem systematically.
### Step 1: Simplify individual components
Let's first simplify each of the components in the expression:
- [tex]\(2^{-3} = 0.125\)[/tex]
- [tex]\(5^{-3} = 0.008\)[/tex]
- [tex]\(10^2 = 100\)[/tex]
- [tex]\(25 = 25\)[/tex]
- [tex]\(5^4 = 625\)[/tex]
- [tex]\(2^{-5} = 0.03125\)[/tex]
### Step 2: Combine the components
Now, substituting these values back into the expression, we get:
[tex]\[
\frac{2^{-3} \times 5^{-3} \times 10^2 \times 25}{5^4 \times 2^{-5}} = \frac{0.125 \times 0.008 \times 100 \times 25}{625 \times 0.03125}
\][/tex]
### Step 3: Calculate the numerator
Let's calculate the value of the numerator:
[tex]\[
0.125 \times 0.008 \times 100 \times 25 = 2.5
\][/tex]
### Step 4: Calculate the denominator
Next, we calculate the value of the denominator:
[tex]\[
625 \times 0.03125 = 19.53125
\][/tex]
### Step 5: Divide the numerator by the denominator
Finally, we divide the numerator by the denominator:
[tex]\[
\frac{2.5}{19.53125} = 0.128
\][/tex]
### Conclusion
Thus, the simplified value of the given expression [tex]\(\frac{2^{-3} \times 5^{-3} \times 10^2 \times 25}{5^4 \times 2^{-5}}\)[/tex] is [tex]\(0.128\)[/tex].
