Simplify :
(i) If `(4^(n+3)xx8^(3-n))/((64^((-n)/(2)))^(2))=2^(9n)xx4^(3n)`, then find the value of 2n.
(ii) If `sqrt(x)+sqrt(x-sqrt(1-x))=1`, then find the value of x.

Options

A. `{:(""(i),(ii)),(3//2,4//5):}`
B. `{:(""(i),(ii)),(2,3//5):}`
C. `{:(""(i),(ii)),(3,16//25):}`
D. `{:(""(i),(ii)),(3,9//25):}`

Answer

To solve the given problems step by step, we will break down each part of the question.

### Part (i)
**Given:**
\[ \frac{4^{(n+3)} \times 8^{(3-n)}}{(64^{(-n/2)})^2} = 2^{9n} \times 4^{3n} \]

**Step 1:** Convert all bases to powers of 2.

- \(4 = 2^2\)
- \(8 = 2^3\)
- \(64 = 2^6\)

Now rewrite the equation:

\[ \frac{(2^2)^{(n+3)} \times (2^3)^{(3-n)}}{((2^6)^{(-n/2)})^2} = 2^{9n} \times (2^2)^{3n} \]

**Step 2:** Simplify the powers.

- The numerator becomes:
\[ (2^2)^{(n+3)} = 2^{2(n+3)} = 2^{2n + 6} \]
\[ (2^3)^{(3-n)} = 2^{3(3-n)} = 2^{9 - 3n} \]

So, the numerator is:
\[ 2^{2n + 6} \times 2^{9 - 3n} = 2^{(2n + 6 + 9 - 3n)} = 2^{(15 - n)} \]

- The denominator becomes:
\[ (2^6)^{(-n/2)} = 2^{-3n} \]
So,
\[ (2^{-3n})^2 = 2^{-6n} \]

Thus, the left-hand side simplifies to:
\[ \frac{2^{15 - n}}{2^{-6n}} = 2^{(15 - n + 6n)} = 2^{(15 + 5n)} \]

**Step 3:** Rewrite the right-hand side.

The right-hand side becomes:
\[ 2^{9n} \times (2^2)^{3n} = 2^{9n} \times 2^{6n} = 2^{(9n + 6n)} = 2^{15n} \]

**Step 4:** Set the exponents equal to each other.

Since the bases are the same, we can equate the exponents:
\[ 15 + 5n = 15n \]

**Step 5:** Solve for \(n\).

Rearranging gives:
\[ 15 = 15n - 5n \]
\[ 15 = 10n \]
\[ n = \frac{15}{10} = \frac{3}{2} \]

**Step 6:** Find \(2n\).

\[ 2n = 2 \times \frac{3}{2} = 3 \]

### Part (ii)
**Given:**
\[ \sqrt{x} + \sqrt{x - \sqrt{1 - x}} = 1 \]

**Step 1:** Rearrange the equation.

Move \(\sqrt{x}\) to the right side:
\[ \sqrt{x - \sqrt{1 - x}} = 1 - \sqrt{x} \]

**Step 2:** Square both sides.

Squaring both sides gives:
\[ x - \sqrt{1 - x} = (1 - \sqrt{x})^2 \]
Expanding the right side:
\[ x - \sqrt{1 - x} = 1 - 2\sqrt{x} + x \]

**Step 3:** Simplify the equation.

Subtract \(x\) from both sides:
\[ -\sqrt{1 - x} = 1 - 2\sqrt{x} \]
Multiply through by -1:
\[ \sqrt{1 - x} = 2\sqrt{x} - 1 \]

**Step 4:** Square both sides again.

Squaring both sides gives:
\[ 1 - x = (2\sqrt{x} - 1)^2 \]
Expanding the right side:
\[ 1 - x = 4x - 4\sqrt{x} + 1 \]
Subtract 1 from both sides:
\[ -x = 4x - 4\sqrt{x} \]
Rearranging gives:
\[ 5x = 4\sqrt{x} \]

**Step 5:** Square both sides again.

Squaring gives:
\[ 25x^2 = 16x \]
Rearranging gives:
\[ 25x^2 - 16x = 0 \]
Factoring out \(x\):
\[ x(25x - 16) = 0 \]

**Step 6:** Solve for \(x\).

This gives:
\[ x = 0 \quad \text{or} \quad 25x - 16 = 0 \Rightarrow x = \frac{16}{25} \]

### Final Answers:
- For part (i), \(2n = 3\).
- For part (ii), \(x = \frac{16}{25}\).