Question

question 3
1 pt
simplify the expression. assume that all variables represent positive real numbers.
\boxed{\dfrac{x^{-6}y^{-4}}{x^{-3}y^{7}}}^{-3/7}
\bigcirc\\ x^{27/7}y^{33/7}
\bigcirc\\ x^{9/7}y^{12}
\bigcirc\\ x^{9/7}y^{33/7}
\bigcirc\\ x^{54/7}y^{33/7}

Explanation:

Step1: Simplify inside the fraction

First, use the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m - n}$. For the $x$-terms: $x^{-6 - (-3)} = x^{-6 + 3} = x^{-3}$. For the $y$-terms: $y^{-4 - 7} = y^{-11}$. So the expression inside the brackets becomes $\frac{x^{-6}y^{-4}}{x^{-3}y^{7}} = x^{-3}y^{-11}$.

Step2: Apply the outer exponent

Now, apply the exponent $-\frac{3}{7}$ to $x^{-3}y^{-11}$ using the power rule $(a^m)^n = a^{m \cdot n}$. For the $x$-term: $(x^{-3})^{-\frac{3}{7}} = x^{(-3) \cdot (-\frac{3}{7})} = x^{\frac{9}{7}}$. For the $y$-term: $(y^{-11})^{-\frac{3}{7}} = y^{(-11) \cdot (-\frac{3}{7})} = y^{\frac{33}{7}}$.

Answer:

C. $x^{9/7}y^{33/7}$ (assuming the third option is labeled C, with the text $x^{9/7}y^{33/7}$)