Sure, let's simplify the given expression step-by-step:
Given expression:
[tex]\[
\frac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}}
\][/tex]
### Step 1: Simplify the constants
Firstly, simplify the constants in the numerator and denominator independently.
[tex]\[
25 \quad \text{and} \quad 5^{-3} \times 10
\][/tex]
Recall that [tex]\(5^{-3} = \frac{1}{5^3} = \frac{1}{125}\)[/tex]. Therefore,
[tex]\[
5^{-3} \times 10 = \frac{1}{125} \times 10 = \frac{10}{125} = \frac{10}{125} = \frac{2}{25} \text{} = 0.4
\][/tex]
Now divide the constant in the numerator, 25, by the constant term [tex]\(0.4\)[/tex]:
[tex]\[
\frac{25}{0.4} = 25 \times 2.5 = 62.5
\][/tex]
In conclusion, the constants simplify to:
[tex]\[
62.5
\][/tex]
### Step 2: Simplify the exponents of [tex]\(t\)[/tex]
Now we proceed with the variable part of the expression. Recall the properties of exponents:
[tex]\[
\frac{t^{-4}}{t^{-8}}
\][/tex]
When dividing terms with the same base, subtract the exponents:
[tex]\[
t^{-4} / t^{-8} = t^{-4 - (-8)} = t^{-4 + 8} = t^{4}
\][/tex]
### Step 3: Combine the results
Now combine the simplified constant part with the simplified variable part:
[tex]\[
62.5 \times t^4
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{62.5 t^4}
\][/tex]
---
Summarizing, we've simplified the given expression to:
[tex]\[
\frac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}} = 62.5 t^4
\][/tex]
