Which expression is equivalent to [tex]\frac{10 x^6 y^{12}}{-5 x^{-2} y^{-6}}[/tex]? Assume [tex]x \neq 0, y \neq 0[/tex].

A. [tex]-50 x^8 y^{18}[/tex]
B. [tex]-2 x^8 y^{18}[/tex]
C. [tex]-2 x^{12} y^{72}[/tex]
D. [tex]5 x^8 y^{18}[/tex]



Answer :

To find the expression equivalent to [tex]\(\frac{10 x^6 y^{12}}{-5 x^{-2} y^{-6}}\)[/tex], let's simplify it step by step.

1. Separate the coefficients and the variables:

[tex]\[ \frac{10 x^6 y^{12}}{-5 x^{-2} y^{-6}} = \frac{10}{-5} \cdot \frac{x^6}{x^{-2}} \cdot \frac{y^{12}}{y^{-6}} \][/tex]

2. Simplify the coefficient part:

[tex]\[ \frac{10}{-5} = -2 \][/tex]

3. Simplify the [tex]\(x\)[/tex] part using the rule [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:

[tex]\[ \frac{x^6}{x^{-2}} = x^{6 - (-2)} = x^{6 + 2} = x^8 \][/tex]

4. Simplify the [tex]\(y\)[/tex] part using the same exponent rule:

[tex]\[ \frac{y^{12}}{y^{-6}} = y^{12 - (-6)} = y^{12 + 6} = y^{18} \][/tex]

Putting everything together, we get:

[tex]\[ -2 \cdot x^8 \cdot y^{18} \][/tex]

Hence, the expression equivalent to [tex]\(\frac{10 x^6 y^{12}}{-5 x^{-2} y^{-6}}\)[/tex] is:

[tex]\[ -2 x^8 y^{18} \][/tex]

So the correct answer is:

[tex]\[ \boxed{-2 x^8 y^{18}} \][/tex]