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Answer :
To simplify the expression [tex]\(\frac{3x^2}{12x^3 - 48x^2}\)[/tex], we will follow these steps:
1. Factor the denominator: First, observe that both terms in the denominator, [tex]\(12x^3\)[/tex] and [tex]\(-48x^2\)[/tex], have a common factor. The greatest common factor (GCF) is [tex]\(12x^2\)[/tex].
2. Rewrite the denominator using the GCF:
[tex]\[
12x^3 - 48x^2 = 12x^2(x - 4)
\][/tex]
3. Rewrite the entire expression using factored form:
[tex]\[
\frac{3x^2}{12x^2(x - 4)}
\][/tex]
4. Simplify the expression by canceling common factors in the numerator and the denominator. Here, [tex]\(3x^2\)[/tex] is a factor in the numerator, and [tex]\(12x^2\)[/tex] is in the denominator, which means:
[tex]\[
\frac{3x^2}{12x^2(x - 4)} = \frac{1}{4(x - 4)}
\][/tex]
The simplified form of the expression is [tex]\(\frac{1}{4(x - 4)}\)[/tex].
1. Factor the denominator: First, observe that both terms in the denominator, [tex]\(12x^3\)[/tex] and [tex]\(-48x^2\)[/tex], have a common factor. The greatest common factor (GCF) is [tex]\(12x^2\)[/tex].
2. Rewrite the denominator using the GCF:
[tex]\[
12x^3 - 48x^2 = 12x^2(x - 4)
\][/tex]
3. Rewrite the entire expression using factored form:
[tex]\[
\frac{3x^2}{12x^2(x - 4)}
\][/tex]
4. Simplify the expression by canceling common factors in the numerator and the denominator. Here, [tex]\(3x^2\)[/tex] is a factor in the numerator, and [tex]\(12x^2\)[/tex] is in the denominator, which means:
[tex]\[
\frac{3x^2}{12x^2(x - 4)} = \frac{1}{4(x - 4)}
\][/tex]
The simplified form of the expression is [tex]\(\frac{1}{4(x - 4)}\)[/tex].
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