Which expression is equivalent to [tex]\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}[/tex]? Assume [tex]x \neq 0, y \neq 0[/tex].

A. [tex]\frac{3}{5 x^5 y^3}[/tex]
B. [tex]\frac{3}{5 x^6 y^6}[/tex]
C. [tex]\frac{5}{3 x^5 y^3}[/tex]
D. [tex]\frac{5}{3 x^6 y^6}[/tex]



Answer :

To determine which expression is equivalent to [tex]\(\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}\)[/tex], follow these steps:

1. Simplify the numerical coefficient:
[tex]\[ \frac{-9}{-15} = \frac{9}{15} = \frac{3}{5} \][/tex]

2. Simplify the [tex]\(x\)[/tex] terms:
- The term in the numerator [tex]\(x^{-1}\)[/tex] can be written as [tex]\(\frac{1}{x}\)[/tex].
- Therefore, the expression has [tex]\(\frac{1}{x}\)[/tex] in the numerator.

- In the denominator, there is [tex]\(x^5\)[/tex].

When simplifying [tex]\(x^{-1}\)[/tex] and [tex]\(x^5\)[/tex], the expression for [tex]\(x\)[/tex] becomes:
[tex]\[ x^{-1} \cdot x^{-5} = x^{-1 - 5} = x^{-6} \][/tex]
Since [tex]\(x^{-6} = \frac{1}{x^6}\)[/tex], we need to recognize that it will be in the denominator as [tex]\(x^6\)[/tex].

3. Simplify the [tex]\(y\)[/tex] terms:
- The numerator has [tex]\(y^{-9}\)[/tex], which is [tex]\(\frac{1}{y^9}\)[/tex].

- The denominator has [tex]\(y^{-3}\)[/tex], which is [tex]\(\frac{1}{y^3}\)[/tex].

When simplifying [tex]\(y^{-9}\)[/tex] and [tex]\(y^{-3}\)[/tex], the expression for [tex]\(y\)[/tex] becomes:
[tex]\[ y^{-9} \cdot y^{3} = y^{-9 + 3} = y^{-6} \][/tex]
Since [tex]\(y^{-6} = \frac{1}{y^6}\)[/tex], we know that [tex]\(y^6\)[/tex] will be in the denominator.

Combining everything, the simplified expression is:
[tex]\[ \frac{3}{5 x^6 y^6} \][/tex]

Hence, the expression equivalent to [tex]\(\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}\)[/tex] is:
[tex]\[ \boxed{\frac{3}{5 x^6 y^6}} \][/tex]