Question

1. simplify ((x^{-2}y^{-3})^{-4})

(x^{-2})
(x^{-8}y^{-12})
(x^{8}y^{-3})
(x^{8}y^{12})

1. simplify ((3x^{-2})^{-2}) and write with positive exponents.

(9x^{-4})
(1/9x^{4})
(9x^{4})
(1/9x^{-4})

Explanation:

Step1: Apply power of a product rule

$(x^{-3}y^{-3})^4 = (x^{-3})^4 \cdot (y^{-3})^4$

Step2: Apply power of a power rule

$(x^{-3})^4 = x^{-3 \times 4} = x^{-12}$; $(y^{-3})^4 = y^{-3 \times 4} = y^{-12}$

Step3: Combine the results

$x^{-12}y^{-12}$

Step1: Apply power of a product rule

$(3x^{-2})^{-2} = 3^{-2} \cdot (x^{-2})^{-2}$

Step2: Simplify constant and exponents

$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$; $(x^{-2})^{-2} = x^{(-2) \times (-2)} = x^{4}$

Step3: Combine the results

$\frac{1}{9}x^4$

Answer:

$x^{-12}y^{-12}$ (matches the second option)

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