If `(3^(a+3) xx 4^(a+6) xx 25^(a+1))/(27^(a-1) xx 8^(a-2) xx 125^(a+4)) = 4/(15^(26))`, then the value of `sqrt(a+9)` is:

Options

A. 8
B. 5
C. 4
D. 6

Answer

To solve the equation

\[ \frac{3^{(a+3)} \cdot 4^{(a+6)} \cdot 25^{(a+1)}}{27^{(a-1)} \cdot 8^{(a-2)} \cdot 125^{(a+4)}} = \frac{4}{15^{26}}, \]

we will first express all the numbers in terms of their prime factors.

### Step 1: Rewrite the bases in terms of prime factors

- \( 4 = 2^2 \)
- \( 25 = 5^2 \)
- \( 27 = 3^3 \)
- \( 8 = 2^3 \)
- \( 125 = 5^3 \)

Now, substituting these into the equation gives:

\[ \frac{3^{(a+3)} \cdot (2^2)^{(a+6)} \cdot (5^2)^{(a+1)}}{(3^3)^{(a-1)} \cdot (2^3)^{(a-2)} \cdot (5^3)^{(a+4)}} = \frac{4}{15^{26}}. \]

### Step 2: Simplify the expression

This simplifies to:

\[ \frac{3^{(a+3)} \cdot 2^{2(a+6)} \cdot 5^{2(a+1)}}{3^{3(a-1)} \cdot 2^{3(a-2)} \cdot 5^{3(a+4)}} = \frac{4}{15^{26}}. \]

Now, simplifying the numerator and denominator:

- For \( 2 \):
- Numerator: \( 2^{2(a+6)} = 2^{2a + 12} \)
- Denominator: \( 2^{3(a-2)} = 2^{3a - 6} \)

Thus, the power of \( 2 \) becomes:

\[ 2^{(2a + 12) - (3a - 6)} = 2^{(2a + 12 - 3a + 6)} = 2^{(18 - a)}. \]

- For \( 3 \):
- Numerator: \( 3^{(a+3)} \)
- Denominator: \( 3^{3(a-1)} = 3^{3a - 3} \)

Thus, the power of \( 3 \) becomes:

\[ 3^{(a + 3) - (3a - 3)} = 3^{(a + 3 - 3a + 3)} = 3^{(6 - 2a)}. \]

- For \( 5 \):
- Numerator: \( 5^{2(a+1)} = 5^{2a + 2} \)
- Denominator: \( 5^{3(a+4)} = 5^{3a + 12} \)

Thus, the power of \( 5 \) becomes:

\[ 5^{(2a + 2) - (3a + 12)} = 5^{(2a + 2 - 3a - 12)} = 5^{(-a - 10)}. \]

### Step 3: Combine the results

Putting it all together, we have:

\[ \frac{2^{(18 - a)} \cdot 3^{(6 - 2a)} \cdot 5^{(-a - 10)}}{1} = \frac{4}{15^{26}}. \]

Now, express \( 4 \) and \( 15^{26} \):

\[ 4 = 2^2, \quad 15^{26} = (3 \cdot 5)^{26} = 3^{26} \cdot 5^{26}. \]

### Step 4: Set up the equation

Thus, we can equate the powers of the bases:

1. For \( 2 \):
\[ 18 - a = 2 \implies a = 16. \]

2. For \( 3 \):
\[ 6 - 2a = 26 \implies -2a = 20 \implies a = -10. \]

3. For \( 5 \):
\[ -a - 10 = 26 \implies -a = 36 \implies a = -36. \]

### Step 5: Find the value of \( \sqrt{a + 9} \)

Since \( a = 16 \) is the only valid solution, we calculate:

\[ \sqrt{a + 9} = \sqrt{16 + 9} = \sqrt{25} = 5. \]

### Final Answer

The value of \( \sqrt{a + 9} \) is \( 5 \).

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