Simplify Exponents 6C5680

1. **State the problem:** Simplify $$\left(\frac{2x^{-3}y^{0}}{4x^{4}y^{-5}}\right)^{-2}$$ and write the answer with positive exponents only. 2. **Recall important rules:** - Any number or variable raised to the zero power is 1, so $$y^{0} = 1$$. - When dividing like bases, subtract exponents: $$\frac{a^{m}}{a^{n}} = a^{m-n}$$. - When raising a fraction to a negative exponent, invert the fraction and change the exponent to positive: $$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$$. - When raising a power to another power, multiply exponents: $$\left(a^{m}\right)^{n} = a^{mn}$$. 3. **Simplify inside the parentheses:** $$\frac{2x^{-3}y^{0}}{4x^{4}y^{-5}} = \frac{2x^{-3} \cdot 1}{4x^{4}y^{-5}} = \frac{2x^{-3}}{4x^{4}y^{-5}}$$ 4. **Simplify coefficients:** $$\frac{2}{4} = \frac{1}{2}$$ 5. **Simplify variables:** $$\frac{x^{-3}}{x^{4}} = x^{-3-4} = x^{-7}$$ $$\frac{1}{y^{-5}} = y^{5}$$ (since dividing by $$y^{-5}$$ is multiplying by $$y^{5}$$) 6. **Combine all:** $$\frac{1}{2} x^{-7} y^{5}$$ 7. **Apply the outer exponent $$-2$$:** $$\left(\frac{1}{2} x^{-7} y^{5}\right)^{-2} = \left(\frac{1}{2}\right)^{-2} \left(x^{-7}\right)^{-2} \left(y^{5}\right)^{-2}$$ 8. **Calculate each part:** $$\left(\frac{1}{2}\right)^{-2} = 2^{2} = 4$$ $$\left(x^{-7}\right)^{-2} = x^{14}$$ $$\left(y^{5}\right)^{-2} = y^{-10}$$ 9. **Combine:** $$4 x^{14} y^{-10}$$ 10. **Rewrite with positive exponents only:** $$4 x^{14} \frac{1}{y^{10}} = \frac{4 x^{14}}{y^{10}}$$ **Final answer:** $$\boxed{\frac{4 x^{14}}{y^{10}}}$$