Let's break down and simplify each expression step-by-step.
### Part (i)
Expression: [tex]\(\frac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}} \quad (t \neq 0)\)[/tex].
1. Simplify the constants:
- The numerator constant: [tex]\(25\)[/tex].
- The denominator constants: [tex]\(5^{-3} \times 10\)[/tex].
To simplify:
[tex]\[
5^{-3} = \frac{1}{5^3} = \frac{1}{125}
\][/tex]
So,
[tex]\[
5^{-3} \times 10 = \frac{1}{125} \times 10 = \frac{10}{125} = \frac{2}{25}
\][/tex]
Thus, the simplified form of constants is:
[tex]\[
\frac{25}{\frac{2}{25}} = 25 \times \frac{25}{2} = \frac{625}{2}
\][/tex]
2. Simplify the powers of [tex]\(t\)[/tex]:
[tex]\[
\frac{t^{-4}}{t^{-8}} = t^{-4 - (-8)} = t^{-4 + 8} = t^{4}
\][/tex]
3. Combine the simplified constants and variable parts:
[tex]\[
\frac{\frac{625}{2}}{1} \cdot t^4 = \frac{625}{2} \times t^4
\][/tex]
So, the final simplified expression for part (i) is:
[tex]\[
\frac{625}{2} \times t^4
\][/tex]
### Part (ii)
Expression: [tex]\(\frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}\)[/tex].
1. Simplify the constants:
- Numerator: [tex]\(3^{-5} \times 10^{-5} \times 125\)[/tex].
- Denominator: [tex]\(5^{-7} \times 6^{-5}\)[/tex].
Let's start with [tex]\(125\)[/tex]:
[tex]\[
125 = 5^3
\][/tex]
Now, let's rewrite the numerator:
[tex]\[
3^{-5} \times 10^{-5} \times 5^3
\][/tex]
For [tex]\(10^{-5}\)[/tex]:
[tex]\[
10^{-5} = (2 \times 5)^{-5} = 2^{-5} \times 5^{-5}
\][/tex]
Thus, the numerator becomes:
[tex]\[
3^{-5} \times 2^{-5} \times 5^{-5} \times 5^3 = 3^{-5} \times 2^{-5} \times 5^{-2}
\][/tex]
Let's rewrite the denominator:
[tex]\[
5^{-7} \times 6^{-5}
\][/tex]
Since [tex]\(6 = 2 \times 3\)[/tex]:
[tex]\[
6^{-5} = (2 \times 3)^{-5} = 2^{-5} \times 3^{-5}
\][/tex]
Thus, the denominator becomes:
[tex]\[
5^{-7} \times 2^{-5} \times 3^{-5}
\][/tex]
2. Combining the parts:
[tex]\[
\frac{3^{-5} \times 2^{-5} \times 5^{-2}}{5^{-7} \times 2^{-5} \times 3^{-5}}
\][/tex]
Notice that [tex]\(3^{-5}\)[/tex] and [tex]\(2^{-5}\)[/tex] cancel out:
[tex]\[
\frac{5^{-2}}{5^{-7}} = 5^{-2 - (-7)} = 5^{-2+7} = 5^5
\][/tex]
So, the final simplified expression for part (ii) is:
[tex]\[
5^5 = 3125
\][/tex]
### Summary
(i) [tex]\(\frac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}} = \frac{625}{2} \times t^4\)[/tex].
(ii) [tex]\(\frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}} = 5^5 = 3125\)[/tex].
