Solutions to Problem Set 10

Math 101-06

9-29-2017

1. Simplify, writing your answer using only positive powers:

$$x^7 \cdot x^{-13}.$$

$$x^7 \cdot x^{-13} = x^{-6} = \dfrac{1}{x^6}.\quad\halmos$$


2. Simplify, writing your answer using only positive powers:

$$(x^{-4})^{-3}.$$

$$(x^{-4})^{-3} = x^{12}.\quad\halmos$$


3. Simplify, writing your answer using only positive powers:

$$\dfrac{x^5}{x^{13}}.$$

$$\dfrac{x^5}{x^{13}} = x^{-8} = \dfrac{1}{x^8}.\quad\halmos$$


4. Simplify, writing your answer using only positive powers:

$$x^6 \cdot (x^{-5})^2.$$

$$x^6 \cdot (x^{-5})^2 = x^6 \cdot x^{-10} = x^{-4} = \dfrac{1}{x^4}.\quad\halmos$$


5. Simplify, writing your answer using only positive powers:

$$\dfrac{x^8}{(x^{-6})^2}.$$

$$\dfrac{x^8}{(x^{-6})^2} = \dfrac{x^8}{x^{-12}} = x^{20}.\quad\halmos$$


6. Simplify, writing your answer using only positive powers:

$$(x^5)^3 \cdot (x^{-4})^2.$$

$$(x^5)^3 \cdot (x^{-4})^2 = x^{15} \cdot x^{-8} = x^7.\quad\halmos$$


7. Simplify, writing your answer using only positive powers:

$$\dfrac{(x^4)^5 \cdot x^{-3}}{(x^{10})^3}.$$

$$\dfrac{(x^4)^5 \cdot x^{-3}}{(x^{10})^3} = \dfrac{x^{20} \cdot x^{-3}}{x^{30}} = \dfrac{x^{17}}{x^{30}} = x^{-13} = \dfrac{1}{x^{13}}.\quad\halmos$$


8. Simplify, writing your answer using only positive powers:

$$(-2 x^4)^3 \cdot 3 (x^{-2})^4.$$

$$(-2 x^4)^3 \cdot 3 (x^{-2})^4 = (-2)^3 (x^4)^3 \cdot 3 (x^{-2})^4 = -8 x^{12} \cdot 3 x^{-8} = -24 x^4.\quad\halmos$$


9. Simplify, writing your answer using only positive powers:

$$\dfrac{3 (x^{-5})^4}{(6 x^7)^2}.$$

$$\dfrac{3 (x^{-5})^4}{(6 x^7)^2} = \dfrac{3 (x^{-5})^4}{6^2 (x^7)^2} = \dfrac{3 x^{-20}}{36 x^{14}} = \dfrac{x^{-34}}{12} = \dfrac{1}{12 x^{34}}.\quad\halmos$$


10. Simplify, writing your answer using only positive powers:

$$\left(\dfrac{(4 x^5)^3}{(6 x^4)^2}\right)^2.$$

$$\left(\dfrac{(4 x^5)^3}{(6 x^4)^2}\right)^2 = \left(\dfrac{4^3 (x^5)^3}{6^2 (x^4)^2}\right)^2 = \left(\dfrac{64 x^{15}}{36 x^8}\right)^2 = \left(\dfrac{16 x^7}{9}\right)^2 = \dfrac{16^2 (x^7)^2}{9^2} = \dfrac{256 x^{14}}{81}.\quad\halmos$$


11. Simplify, writing your answer using only positive powers:

$$(6 x^2 y^{-3})^2 \cdot 5 (x^{-7} y^4)^3.$$

$$(6 x^2 y^{-3})^2 \cdot 5 (x^{-7} y^4)^3 = 6^2 (x^2)^2 (y^{-3})^2 \cdot 5 (x^{-7})^3 (y^4)^3 = 36 x^4 y^{-6} \cdot 5 x^{-21} y^{12} = 180 x^{-17} y^6 = \dfrac{180 y^6}{x^{17}}.\quad\halmos$$


12. Simplify, writing your answer using only positive powers:

$$\dfrac{(4 x^5 y^{-7})^2}{12 (x^{-3} y)^3}.$$

$$\dfrac{(4 x^5 y^{-7})^2}{12 (x^{-3} y)^3} = \dfrac{4^2 (x^5)^2 (y^{-7})^2}{12 (x^{-3})^3 y^3} = \dfrac{16 x^{10} y^{-14}}{12 x^{-9} y^3} = \dfrac{4 x^{19} y^{-17}}{3} = \dfrac{4 x^{19}}{3 y^{17}}.\quad\halmos$$


13. Simplify, writing your answer using only positive powers:

$$\left(\dfrac{27 (x^5 y^{-1})^3}{(3 x^6 y^4)^2}\right)^2.$$

$$\left(\dfrac{27 (x^5 y^{-1})^3}{(3 x^6 y^4)^2}\right)^2 = \left(\dfrac{27 (x^5)^3 (y^{-1})^3}{3^2 (x^6)^2 (y^4)^2}\right)^2 = \left(\dfrac{27 x^{15} y^{-3}}{9 x^{12} y^8}\right)^2 = \left(3 x^3 y^{-11}\right)^2 =$$

$$3^2 (x^3)^2 (y^{-11})^2 = 9 x^6 y^{-22} = \dfrac{9 x^6}{y^{22}}.\quad\halmos$$


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