If `(5^(8)xx6^(7))/((25)^(3)xx6^(4))= (3125)^(b)xx a^(3)`, then find the value of a + b.

Options

A. `(25)/(3)`
B. `(3)/(5)`
C. `6(2)/(5)`
D. `(5)/(3)`

Answer

To solve the equation \(\frac{5^8 \times 6^7}{25^3 \times 6^4} = 3125^b \times a^3\), we will simplify both sides step by step.

### Step 1: Rewrite \(25\) and \(3125\) in terms of \(5\)

We know that:
- \(25 = 5^2\)
- \(3125 = 5^5\)

So we can rewrite the equation as:
\[ \frac{5^8 \times 6^7}{(5^2)^3 \times 6^4} = (5^5)^b \times a^3 \]

### Step 2: Simplify the left side

Now, simplify the left side:
\[ (5^2)^3 = 5^{2 \times 3} = 5^6 \]
Thus, the left side becomes:
\[ \frac{5^8 \times 6^7}{5^6 \times 6^4} \]

### Step 3: Apply the laws of exponents

Using the laws of exponents, we can simplify further:
\[ = 5^{8-6} \times \frac{6^7}{6^4} = 5^2 \times 6^{7-4} = 5^2 \times 6^3 \]

### Step 4: Rewrite the right side

Now, rewrite the right side:
\[ (5^5)^b = 5^{5b} \]
So the right side becomes:
\[ 5^{5b} \times a^3 \]

### Step 5: Set the bases equal

Now, we have:
\[ 5^2 \times 6^3 = 5^{5b} \times a^3 \]

From this, we can equate the powers of \(5\) and \(6\):
1. For \(5\): \(2 = 5b\)
2. For \(6\): \(3 = 3\) (which is always true)

### Step 6: Solve for \(b\)

From \(2 = 5b\):
\[ b = \frac{2}{5} \]

### Step 7: Solve for \(a\)

Since \(6^3 = a^3\), we can find \(a\) by taking the cube root:
\[ a = 6 \]

### Step 8: Find \(a + b\)

Now we can find \(a + b\):
\[ a + b = 6 + \frac{2}{5} \]

To add these, convert \(6\) to a fraction:
\[ 6 = \frac{30}{5} \]
Thus,
\[ a + b = \frac{30}{5} + \frac{2}{5} = \frac{32}{5} \]

### Final Answer

The value of \(a + b\) is \(\frac{32}{5}\).

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