Answer :

Sure! Let's simplify the expression [tex]\(\frac{3x^2}{12x^3 - 48x^2}\)[/tex] step by step.

### Step 1: Factor the denominator

The expression in the denominator is [tex]\(12x^3 - 48x^2\)[/tex]. We can factor out the greatest common factor (GCF) first:

1. Identify the GCF of the terms in the denominator:
- The GCF of the coefficients 12 and 48 is 12.
- The smallest power of [tex]\(x\)[/tex] that appears in all terms is [tex]\(x^2\)[/tex].

2. Factor out [tex]\(12x^2\)[/tex]:
[tex]\[
12x^3 - 48x^2 = 12x^2(x - 4)
\][/tex]

### Step 2: Rewrite the expression

Now substitute the factored form of the denominator back into the original expression:
[tex]\[
\frac{3x^2}{12x^2(x - 4)}
\][/tex]

### Step 3: Simplify the expression

Next, simplify the fraction by canceling out the common factors from the numerator and the denominator:

1. Both the numerator [tex]\(3x^2\)[/tex] and the denominator [tex]\(12x^2(x-4)\)[/tex] have a common factor of [tex]\(x^2\)[/tex].

2. Cancel out [tex]\(x^2\)[/tex]:
[tex]\[
\frac{3x^2}{12x^2(x - 4)} = \frac{3}{12(x - 4)}
\][/tex]

3. Simplify the constant fraction [tex]\(\frac{3}{12}\)[/tex]:
- The GCF of 3 and 12 is 3.
- Divide both the numerator and the denominator by 3:
[tex]\[
\frac{3}{12} = \frac{1}{4}
\][/tex]

4. The simplified expression is:
[tex]\[
\frac{1}{4(x - 4)}
\][/tex]

So, the simplified form of the expression [tex]\(\frac{3x^2}{12x^3 - 48x^2}\)[/tex] is [tex]\(\frac{1}{4(x - 4)}\)[/tex].

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