Which expression is equivalent to [tex]\frac{\left(x^6 y^8\right)^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 \(\frac{(x^6 y^8)^3}{x^2 y^2}\), let's go through the problem step by step:

1. Start with the given expression:
[tex]\[ \frac{(x^6 y^8)^3}{x^2 y^2} \][/tex]

2. Simplify the numerator using the power rule \((a^m)^n = a^{m \cdot n}\):
[tex]\[ (x^6 y^8)^3 = x^{6 \cdot 3} y^{8 \cdot 3} = x^{18} y^{24} \][/tex]

3. Rewrite the expression with the simplified numerator:
[tex]\[ \frac{x^{18} y^{24}}{x^2 y^2} \][/tex]

4. Use the quotient rule for exponents, which states \(\frac{a^m}{a^n} = a^{m-n}\), to simplify the expression:
[tex]\[ \frac{x^{18}}{x^2} = x^{18-2} = x^{16} \][/tex]
[tex]\[ \frac{y^{24}}{y^2} = y^{24-2} = y^{22} \][/tex]

5. Combine the results:
[tex]\[ \frac{x^{18} y^{24}}{x^2 y^2} = x^{16} y^{22} \][/tex]

Therefore, the equivalent expression is:

[tex]\(\boxed{x^{16} y^{22}}\)[/tex]