Simplify :
(i) `(root(5)(a^3b^(-2)c^(-4)))/(root5(a^8b^3c^(-9)))` (ii) `(125)^(1/4)(8^(1/3)+27^(1/3))^(1/4)`

Options

A. `{:("(i)","(ii)"),((ab)/c,1/5):}`
B. `{:("(i)","(ii)"),((c)/(ab),5):}`
C. `{:("(i)","(ii)"),((a)/(bc),25):}`
D. `{:("(i)","(ii)"),(abc,30):}`

Answer

Let's simplify the given expressions step by step.

### Part (i)
We need to simplify:
\[ \frac{\sqrt[5]{a^3 b^{-2} c^{-4}}}{\sqrt[5]{a^8 b^3 c^{-9}}} \]

**Step 1: Rewrite the expression using exponents.**
The expression can be rewritten as:
\[ \frac{(a^3 b^{-2} c^{-4})^{1/5}}{(a^8 b^3 c^{-9})^{1/5}} \]

**Step 2: Apply the exponent to each term in the numerator and denominator.**
This gives us:
\[ \frac{a^{3/5} b^{-2/5} c^{-4/5}}{a^{8/5} b^{3/5} c^{-9/5}} \]

**Step 3: Use the property of exponents for division.**
Recall that \(\frac{A^m}{A^n} = A^{m-n}\). Thus, we can simplify each base:
- For \(a\):
\[ a^{3/5 - 8/5} = a^{-5/5} = a^{-1} \]
- For \(b\):
\[ b^{-2/5 - 3/5} = b^{-5/5} = b^{-1} \]
- For \(c\):
\[ c^{-4/5 - (-9/5)} = c^{-4/5 + 9/5} = c^{5/5} = c^{1} \]

**Step 4: Combine the results.**
Putting it all together, we have:
\[ \frac{c}{a b} \]

### Final Answer for Part (i):
\[ \frac{c}{ab} \]

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### Part (ii)
We need to simplify:
\[ (125)^{1/4} \left(8^{1/3} + 27^{1/3}\right)^{1/4} \]

**Step 1: Simplify \(125^{1/4}\).**
We know that \(125 = 5^3\), so:
\[ (125)^{1/4} = (5^3)^{1/4} = 5^{3/4} \]

**Step 2: Simplify \(8^{1/3}\) and \(27^{1/3}\).**
We know that \(8 = 2^3\) and \(27 = 3^3\):
\[ 8^{1/3} = (2^3)^{1/3} = 2 \]
\[ 27^{1/3} = (3^3)^{1/3} = 3 \]

**Step 3: Combine the results inside the parentheses.**
Thus:
\[ 8^{1/3} + 27^{1/3} = 2 + 3 = 5 \]

**Step 4: Apply the exponent to the sum.**
Now we have:
\[ (5)^{1/4} \]

**Step 5: Combine everything.**
Putting it all together:
\[ 5^{3/4} \cdot (5^{1/4}) = 5^{3/4 + 1/4} = 5^{4/4} = 5^1 = 5 \]

### Final Answer for Part (ii):
\[ 5 \]

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