Find the value of n when:
( i ) `5^(2n) xx 5^(3)=5^(9)`
( ii ) `8 xx 2^(n+2)=32`
( iii ) `6^(2n+1) div 36=6^(3)`

Answer

Let's solve the given questions step by step.

### Part (i):
**Equation:** \( 5^{2n} \times 5^{3} = 5^{9} \)

**Step 1:** Use the property of exponents that states \( a^m \times a^n = a^{m+n} \).

\[ 5^{2n + 3} = 5^{9} \]

**Step 2:** Since the bases are the same, we can equate the exponents:

\[ 2n + 3 = 9 \]

**Step 3:** Solve for \( n \) by isolating it. Subtract 3 from both sides:

\[ 2n = 9 - 3 \]
\[ 2n = 6 \]

**Step 4:** Divide both sides by 2:

\[ n = \frac{6}{2} \]
\[ n = 3 \]

### Part (ii):
**Equation:** \( 8 \times 2^{n+2} = 32 \)

**Step 1:** Rewrite 8 and 32 as powers of 2. We know that \( 8 = 2^3 \) and \( 32 = 2^5 \):

\[ 2^3 \times 2^{n+2} = 2^5 \]

**Step 2:** Again, use the property of exponents:

\[ 2^{3 + n + 2} = 2^5 \]
\[ 2^{n + 5} = 2^5 \]

**Step 3:** Equate the exponents since the bases are the same:

\[ n + 5 = 5 \]

**Step 4:** Isolate \( n \) by subtracting 5 from both sides:

\[ n = 5 - 5 \]
\[ n = 0 \]

### Part (iii):
**Equation:** \( \frac{6^{2n+1}}{36} = 6^{3} \)

**Step 1:** Rewrite 36 as \( 6^2 \):

\[ \frac{6^{2n+1}}{6^2} = 6^3 \]

**Step 2:** Use the property of exponents for division \( \frac{a^m}{a^n} = a^{m-n} \):

\[ 6^{(2n + 1) - 2} = 6^3 \]
\[ 6^{2n - 1} = 6^3 \]

**Step 3:** Equate the exponents:

\[ 2n - 1 = 3 \]

**Step 4:** Solve for \( n \) by adding 1 to both sides:

\[ 2n = 3 + 1 \]
\[ 2n = 4 \]

**Step 5:** Divide both sides by 2:

\[ n = \frac{4}{2} \]
\[ n = 2 \]

### Summary of Solutions:
- (i) \( n = 3 \)
- (ii) \( n = 0 \)
- (iii) \( n = 2 \)