Which expression is equivalent to [tex]\frac{(x^6 y^8)^3}{x^2 y^2}[/tex]?

A. [tex]x^7 y^9[/tex]
B. [tex]x^9 y^{12}[/tex]
C. [tex]x^{12} y^{18}[/tex]
D. [tex]x^{16} y^{22}[/tex]



Answer :

To determine which expression is equivalent to [tex]\(\frac{(x^6 y^8)^3}{x^2 y^2}\)[/tex], let's break it down step by step.

1. Simplifying the Numerator:
[tex]\[ (x^6 y^8)^3 \][/tex]
Using the power of a power property [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[ x^{6 \cdot 3} \cdot y^{8 \cdot 3} = x^{18} \cdot y^{24} \][/tex]

2. Simplifying the Entire Fraction:
Substituting the simplified numerator back into the fraction:
[tex]\[ \frac{x^{18} y^{24}}{x^2 y^2} \][/tex]
Using the division property of exponents [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]:
[tex]\[ \frac{x^{18}}{x^2} \cdot \frac{y^{24}}{y^2} = x^{18-2} \cdot y^{24-2} = x^{16} \cdot y^{22} \][/tex]

Thus, the equivalent expression is [tex]\(x^{16} y^{22}\)[/tex].

Therefore, the correct answer is:
[tex]\[ \boxed{x^{16} y^{22}} \][/tex]

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