To solve this problem, we'll substitute the given expressions for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] and simplify the expression [tex]\(\frac{ab}{c}\)[/tex]. Let's take a detailed step-by-step approach:
1. Substitute the expressions for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
a = 4x^3y^2
\][/tex]
[tex]\[
b = 3x^2y^3
\][/tex]
[tex]\[
c = 6xy
\][/tex]
2. Find the product [tex]\(ab\)[/tex]:
First, multiply [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
ab = (4x^3y^2) \cdot (3x^2y^3)
\][/tex]
Multiply the coefficients and then the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
ab = 12x^{3+2}y^{2+3} = 12x^5y^5
\][/tex]
3. Divide the product [tex]\(ab\)[/tex] by [tex]\(c\)[/tex]:
Substitute [tex]\(ab\)[/tex] and [tex]\(c\)[/tex] into the expression [tex]\(\frac{ab}{c}\)[/tex]:
[tex]\[
\frac{ab}{c} = \frac{12x^5y^5}{6xy}
\][/tex]
4. Simplify the division:
First, divide the coefficients:
[tex]\[
\frac{12}{6} = 2
\][/tex]
Then, apply the properties of exponents to divide [tex]\(x^5\)[/tex] by [tex]\(x\)[/tex] and [tex]\(y^5\)[/tex] by [tex]\(y\)[/tex]:
[tex]\[
\frac{x^5}{x} = x^{5-1} = x^4
\][/tex]
[tex]\[
\frac{y^5}{y} = y^{5-1} = y^4
\][/tex]
So,
[tex]\[
\frac{12x^5y^5}{6xy} = 2x^4y^4
\][/tex]
5. Write the final simplified result:
[tex]\[
\frac{ab}{c} = 2x^4y^4
\][/tex]
Hence, the value of [tex]\(\frac{ab}{c}\)[/tex] is [tex]\(2x^4y^4\)[/tex].
