$$ \frac { 2 ^ { 3 b + 2 } \cdot 7 ^ { 3 b - 2 } } { 14 ^ { 3 b } - 14 ^ { 3 b - 1 } \cdot 2 } $$

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UpStudy AI · Accepted Answer

Let's simplify the expression:

$$
\frac{2^{3b + 2} \cdot 7^{3b - 2}}{14^{3b} - 14^{3b - 1} \cdot 2}
$$

**Step 1: Express Everything in Terms of 2 and 7**

Since $$14 = 2 \times 7$$, we can rewrite the denominator:

$$
14^{3b} = (2 \times 7)^{3b} = 2^{3b} \cdot 7^{3b}
$$
$$
14^{3b - 1} = (2 \times 7)^{3b - 1} = 2^{3b - 1} \cdot 7^{3b - 1}
$$

So, the denominator becomes:

$$
14^{3b} - 2 \cdot 14^{3b - 1} = 2^{3b} \cdot 7^{3b} - 2 \cdot (2^{3b - 1} \cdot 7^{3b - 1})
$$

**Step 2: Simplify the Denominator**

Factor out $$2^{3b} \cdot 7^{3b}$$:

$$
2^{3b} \cdot 7^{3b} - 2^{3b} \cdot 7^{3b} \cdot \frac{2}{2} \cdot \frac{1}{7} = 2^{3b} \cdot 7^{3b} \left(1 - \frac{1}{7}\right) = 2^{3b} \cdot 7^{3b} \cdot \frac{6}{7}
$$

So, the denominator simplifies to:

$$
\frac{6}{7} \cdot 2^{3b} \cdot 7^{3b}
$$

**Step 3: Simplify the Entire Expression**

Now, the original expression becomes:

$$
\frac{2^{3b + 2} \cdot 7^{3b - 2}}{\frac{6}{7} \cdot 2^{3b} \cdot 7^{3b}} = \frac{2^2 \cdot 7^{-2}}{\frac{6}{7}} = \frac{4}{49} \cdot \frac{7}{6} = \frac{4 \cdot 7}{49 \cdot 6} = \frac{28}{294}
$$

**Step 4: Reduce the Fraction**

$$
\frac{28}{294} = \frac{2}{21}
$$

**Final Answer:**

$$
\frac{2}{21}
$$
The simplified form of the expression is $$ \frac{2}{21} $$.

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