Q:

# Which expression is a perfect cube? A. -8x21y8B. -64x64y64C. -125x9y20D. -216x9y18

Accepted Solution

A:
Finding the cube root of each expression

Expression A:

$$\sqrt[3]{-8x^{21}y^{8}}$$
$$\sqrt[3]{-8}$$ $$\sqrt[3]{x^{21}[tex] \sqrt[3]{y^8}$$} [/tex]
$$-2x^ \frac{21}{3} y^ \frac{8}{3}$$
$$-2 x^7 y^ \frac{8}{3}$$

the term $$y^ \frac{8}{3}$$ is not a perfect cube, therefore expression A is not a perfect cube.

--------------------------------------------------------------------------------------------------------------

Expression B
$$\sqrt[3]{-64x^{64}y^{64}}$$
$$\sqrt[3]{-64} \sqrt[3]{x^{64}} \sqrt[3]{y^{64}}$$
$$-4 x^ \frac{64}{3} y^ \frac{64}{3}$$

The term $$x^ \frac{64}{3}$$ and $$y^ \frac{64}{3}$$ are not perfect cube because 64/3 doesn't give whole number
----------------------------------------------------------------------------------------------------------------

Expression C

$$\sqrt[3] {-125x^9y^{20}}$$
$$\sqrt[3]{-125} \sqrt[3]{x^9} \sqrt[3] {y^{20}}$$
$$-5 (x^ \frac{9}{3}) (y^ \frac{20}{3})$$
$$-5 (x^3) (y^ \frac{20} {3})$$

The term $$y^ \frac{20}{3}$$ are not perfect cube because 20/3 doesn't give whole number
---------------------------------------------------------------------------------------------------------------

Expression D

$$\sqrt[3]{ -216 x^{9} y^{18} }$$
$$\sqrt[3]{-216} \sqrt[3]{x^9} \sqrt[3]{y^{18}}$$
$$-6 ( x^ \frac{9} {3} ) ( y^ \frac{18} {3} )$$
$$-6 (x^3) (y^6)$$

All terms are perfect cubes