`7/(12b^3)-3/(4b^3)`
The above expression is equivalent to which of the following expressions for all `b gt 0`?

Options

A. `-1/(6b^3)`
B. `-1/(4b^3)`
C. `1/(4b^3)`
D. `1/(6b^3)`

Answer

To solve the expression \( \frac{7}{12b^3} - \frac{3}{4b^3} \) and find its equivalent expression for all \( b > 0 \), we can follow these steps:

### Step 1: Identify a common denominator
The denominators in the expression are \( 12b^3 \) and \( 4b^3 \). The least common multiple (LCM) of \( 12 \) and \( 4 \) is \( 12 \). Thus, we will use \( 12b^3 \) as the common denominator.

### Step 2: Rewrite each fraction with the common denominator
The first fraction is already in terms of the common denominator:
\[ \frac{7}{12b^3} \]
For the second fraction, we need to adjust it:
\[ \frac{3}{4b^3} = \frac{3 \cdot 3}{4b^3 \cdot 3} = \frac{9}{12b^3} \]

### Step 3: Combine the fractions
Now we can combine the two fractions:
\[ \frac{7}{12b^3} - \frac{9}{12b^3} = \frac{7 - 9}{12b^3} = \frac{-2}{12b^3} \]

### Step 4: Simplify the expression
Next, we simplify \( \frac{-2}{12b^3} \):
\[ \frac{-2}{12b^3} = \frac{-1}{6b^3} \]

### Step 5: Write the final expression
Thus, the expression \( \frac{7}{12b^3} - \frac{3}{4b^3} \) simplifies to:
\[ -\frac{1}{6b^3} \]

### Conclusion
The equivalent expression for \( \frac{7}{12b^3} - \frac{3}{4b^3} \) for all \( b > 0 \) is:
\[ -\frac{1}{6b^3} \]