If `(a^(2b-3)xx(a^(2))^(b+1))/((a^(4))^(-3))=(a^(3))^(3)-:(a^(6))^(-3)`, then find the value of 2b.

Options

A. 4
B. 8
C. 16
D. 24

Answer

To solve the equation

\[ \frac{a^{(2b-3) \cdot (a^2)^{(b+1)}}}{(a^4)^{-3}} = \frac{(a^3)}{(a^6)^{-3}}, \]

we will simplify both sides step by step.

### Step 1: Simplify the left-hand side

The left-hand side can be simplified as follows:

1. **Simplify the exponent on \( (a^2)^{(b+1)} \)**:
\[ (a^2)^{(b+1)} = a^{2(b+1)} = a^{2b + 2} \]

2. **Combine the exponents in the numerator**:
\[ a^{(2b-3) + (2b + 2)} = a^{(2b - 3 + 2b + 2)} = a^{4b - 1} \]

3. **Simplify the denominator**:
\[ (a^4)^{-3} = a^{-12} \]

4. **Combine the numerator and denominator**:
\[ \frac{a^{4b - 1}}{a^{-12}} = a^{(4b - 1) - (-12)} = a^{4b - 1 + 12} = a^{4b + 11} \]

### Step 2: Simplify the right-hand side

Now, simplify the right-hand side:

1. **Simplify the exponent on \( (a^6)^{-3} \)**:
\[ (a^6)^{-3} = a^{-18} \]

2. **Combine the terms**:
\[ \frac{a^3}{a^{-18}} = a^{3 - (-18)} = a^{3 + 18} = a^{21} \]

### Step 3: Set the simplified expressions equal to each other

Now we have:

\[ a^{4b + 11} = a^{21} \]

### Step 4: Set the exponents equal to each other

Since the bases are the same, we can set the exponents equal:

\[ 4b + 11 = 21 \]

### Step 5: Solve for \( b \)

1. **Subtract 11 from both sides**:
\[ 4b = 21 - 11 \]
\[ 4b = 10 \]

2. **Divide by 4**:
\[ b = \frac{10}{4} = 2.5 \]

### Step 6: Find \( 2b \)

Now, we need to find \( 2b \):
\[ 2b = 2 \times 2.5 = 5 \]

Thus, the value of \( 2b \) is **5**.

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