Question 1 of 5: Simplify, assume all variables represent positive real numbers.
Select the Correct Answer: Correct! 
Not Correct! 
Not Correct! A
$$\sqrt{45}=9\sqrt{5},\sqrt{27}=9\sqrt{3},8\sqrt[4]{32x^9}=4x^2\sqrt[4]{4x}$$ $$\sqrt{45}=9\sqrt{5}$$$$\sqrt{27}=9\sqrt{3}$$$$8\sqrt[4]{32x^9}=4x^2\sqrt[4]{4x}$$ $$\sqrt{45}=9\sqrt{5}$$$$\sqrt{27}=9\sqrt{3}$$$$8\sqrt[4]{32x^9}=$$$$4x^2\sqrt[4]{4x}$$
B
$$\sqrt{45}=5\sqrt{3},\sqrt{27}=9\sqrt{3},8\sqrt[4]{32x^9}=8x^2\sqrt[4]{8x}$$ $$\sqrt{45}=5\sqrt{3}$$$$\sqrt{27}=9\sqrt{3}$$$$8\sqrt[4]{32x^9}=8x^2\sqrt[4]{8x}$$ $$\sqrt{45}=5\sqrt{3}$$$$\sqrt{27}=9\sqrt{3}$$$$8\sqrt[4]{32x^9}=$$$$8x^2\sqrt[4]{8x}$$
C
$$\sqrt{45}=3\sqrt{5},\sqrt{27}=3\sqrt{3},8\sqrt[4]{32x^9}=16x^2\sqrt[4]{2x}$$ $$\sqrt{45}=3\sqrt{5}$$$$\sqrt{27}=3\sqrt{3}$$$$8\sqrt[4]{32x^9}=16x^2\sqrt[4]{2x}$$ $$\sqrt{45}=3\sqrt{5}$$$$\sqrt{27}=3\sqrt{3}$$$$8\sqrt[4]{32x^9}=$$$$16x^2\sqrt[4]{2x}$$
D
$$\sqrt{45}=\sqrt{5},\sqrt{27}=\sqrt{3},8\sqrt[4]{32x^9}=\sqrt[4]{2x}$$ $$\sqrt{45}=\sqrt{5}$$$$\sqrt{27}=\sqrt{3}$$$$8\sqrt[4]{32x^9}=\sqrt[4]{2x}$$ $$\sqrt{45}=\sqrt{5}$$$$\sqrt{27}=\sqrt{3}$$$$8\sqrt[4]{32x^9}=$$$$\sqrt[4]{2x}$$
E
$$\sqrt{45}=3\sqrt{5},\sqrt{27}=9\sqrt{3},8\sqrt[4]{32x^9}=4x^2\sqrt[4]{x}$$ $$\sqrt{45}=3\sqrt{5}$$$$\sqrt{27}=9\sqrt{3}$$$$8\sqrt[4]{32x^9}=4x^2\sqrt[4]{x}$$ $$\sqrt{45}=3\sqrt{5}$$$$\sqrt{27}=9\sqrt{3}$$$$8\sqrt[4]{32x^9}=$$$$4x^2\sqrt[4]{x}$$
Question 2 of 5: Simplify, assume all variables represent positive real numbers.
Select the Correct Answer: Correct! Not Correct!
Not Correct! A
$$-4\sqrt{2}\cdot3\sqrt{8}=-2\sqrt{2},\sqrt{5}\cdot\sqrt{10}=25\sqrt{2}, \sqrt[3]{-9x^4}\cdot\sqrt[3]{-45x^3}=9x^2\sqrt[3]{5x}$$ $$-4\sqrt{2}\cdot3\sqrt{8}=-2\sqrt{2}$$$$\sqrt{5}\cdot\sqrt{10}=25\sqrt{2}$$$$ \sqrt[3]{-9x^4}\cdot\sqrt[3]{-45x^3}=9x^2\sqrt[3]{5x}$$ $$-4\sqrt{2}\cdot3\sqrt{8}$$$$=-2\sqrt{2}$$$$\sqrt{5}\cdot\sqrt{10}$$$$=25\sqrt{2}$$$$ \sqrt[3]{-9x^4}\hspace{.25em}\cdot$$$$\sqrt[3]{-45x^3}=$$$$9x^2\sqrt[3]{5x}$$
B
$$-4\sqrt{2}\cdot3\sqrt{8}=-48,\sqrt{5}\cdot\sqrt{10}=5\sqrt{2}, \sqrt[3]{-9x^4}\cdot\sqrt[3]{-45x^3}=3x^2\sqrt[3]{15x}$$ $$-4\sqrt{2}\cdot3\sqrt{8}=-48$$$$\sqrt{5}\cdot\sqrt{10}=5\sqrt{2}$$$$ \sqrt[3]{-9x^4}\cdot\sqrt[3]{-45x^3}=3x^2\sqrt[3]{15x}$$ $$-4\sqrt{2}\cdot3\sqrt{8}$$$$=-48$$$$\sqrt{5}\cdot\sqrt{10}$$$$=5\sqrt{2}$$$$ \sqrt[3]{-9x^4}\hspace{.25em}\cdot$$$$\sqrt[3]{-45x^3}$$$$=3x^2\sqrt[3]{15x}$$
C
$$-4\sqrt{2}\cdot3\sqrt{8}=-6\sqrt{8},\sqrt{5}\cdot\sqrt{10}=5\sqrt{5}, \sqrt[3]{-9x^4}\cdot\sqrt[3]{-45x^3}=6x\sqrt[3]{15x}$$ $$-4\sqrt{2}\cdot3\sqrt{8}=-6\sqrt{8}$$$$\sqrt{5}\cdot\sqrt{10}=5\sqrt{5}$$$$ \sqrt[3]{-9x^4}\cdot\sqrt[3]{-45x^3}=6x\sqrt[3]{15x}$$ $$-4\sqrt{2}\cdot3\sqrt{8}$$$$=-6\sqrt{8}$$$$\sqrt{5}\cdot\sqrt{10}$$$$=5\sqrt{5}$$$$ \sqrt[3]{-9x^4}\hspace{.25em}\cdot$$$$\sqrt[3]{-45x^3}$$$$=6x\sqrt[3]{15x}$$
D
$$-4\sqrt{2}\cdot3\sqrt{8}=-2\sqrt{6},\sqrt{5}\cdot\sqrt{10}=2\sqrt{5}, \sqrt[3]{-9x^4}\cdot\sqrt[3]{-45x^3}=x^2\sqrt[3]{9x}$$ $$-4\sqrt{2}\cdot3\sqrt{8}=-2\sqrt{6}$$$$\sqrt{5}\cdot\sqrt{10}=2\sqrt{5}$$$$ \sqrt[3]{-9x^4}\cdot\sqrt[3]{-45x^3}=x^2\sqrt[3]{9x}$$ $$-4\sqrt{2}\cdot3\sqrt{8}$$$$=-2\sqrt{6}$$$$\sqrt{5}\cdot\sqrt{10}$$$$=2\sqrt{5}$$$$ \sqrt[3]{-9x^4}\hspace{.25em}\cdot$$$$\sqrt[3]{-45x^3}$$$$=x^2\sqrt[3]{9x}$$
E
$$-4\sqrt{2}\cdot3\sqrt{8}=-16,\sqrt{5}\cdot\sqrt{10}=5\sqrt{2}, \sqrt[3]{-9x^4}\cdot\sqrt[3]{-45x^3}=9x^2\sqrt[3]{5x}$$ $$-4\sqrt{2}\cdot3\sqrt{8}=-16$$$$\sqrt{5}\cdot\sqrt{10}=5\sqrt{2}$$$$ \sqrt[3]{-9x^4}\cdot\sqrt[3]{-45x^3}=9x^2\sqrt[3]{5x}$$ $$-4\sqrt{2}\cdot3\sqrt{8}$$$$=-16$$$$\sqrt{5}\cdot\sqrt{10}$$$$=5\sqrt{2}$$$$ \sqrt[3]{-9x^4}\hspace{.25em}\cdot$$$$\sqrt[3]{-45x^3}=$$$$9x^2\sqrt[3]{5x}$$
Question 3 of 5: Simplify, assume all variables represent positive real numbers.
Select the Correct Answer: Correct! Not Correct!
Not Correct! A
$$-6\sqrt{42x^3}\cdot-8\sqrt{28x^2}=169x^2\sqrt{2x}, 4\sqrt[4]{4x^5}\cdot-8\sqrt[4]{3x^3}=-8x^2\sqrt[4]{8}$$ $$-6\sqrt{42x^3}\cdot-8\sqrt{28x^2}=169x^2\sqrt{2x}$$$$ 4\sqrt[4]{4x^5}\cdot-8\sqrt[4]{3x^3}=-8x^2\sqrt[4]{8}$$ $$-6\sqrt{42x^3}\hspace{.25em}\cdot$$$$-8\sqrt{28x^2}$$$$=169x^2\sqrt{2x}$$$$ 4\sqrt[4]{4x^5}\hspace{.25em}\cdot$$$$-8\sqrt[4]{3x^3}$$$$=-8x^2\sqrt[4]{8}$$
B
$$-6\sqrt{42x^3}\cdot-8\sqrt{28x^2}=672x^2\sqrt{6x}, 4\sqrt[4]{4x^5}\cdot-8\sqrt[4]{3x^3}=-32x^2\sqrt[4]{12}$$ $$-6\sqrt{42x^3}\cdot-8\sqrt{28x^2}=672x^2\sqrt{6x}$$$$ 4\sqrt[4]{4x^5}\cdot-8\sqrt[4]{3x^3}=-32x^2\sqrt[4]{12}$$ $$-6\sqrt{42x^3}\hspace{.25em}\cdot$$$$-8\sqrt{28x^2}$$$$=672x^2\sqrt{6x}$$$$ 4\sqrt[4]{4x^5}\hspace{.25em}\cdot$$$$-8\sqrt[4]{3x^3}$$$$=-32x^2\sqrt[4]{12}$$
C
$$-6\sqrt{42x^3}\cdot-8\sqrt{28x^2}=225x\sqrt{10x}, 4\sqrt[4]{4x^5}\cdot-8\sqrt[4]{3x^3}=-2x^2\sqrt[4]{8}$$ $$-6\sqrt{42x^3}\cdot-8\sqrt{28x^2}=225x\sqrt{10x}$$$$ 4\sqrt[4]{4x^5}\cdot-8\sqrt[4]{3x^3}=-2x^2\sqrt[4]{8}$$ $$-6\sqrt{42x^3}\hspace{.25em}\cdot$$$$-8\sqrt{28x^2}$$$$=225x\sqrt{10x}$$$$ 4\sqrt[4]{4x^5}\hspace{.25em}\cdot$$$$-8\sqrt[4]{3x^3}$$$$=-2x^2\sqrt[4]{8}$$
D
$$-6\sqrt{42x^3}\cdot-8\sqrt{28x^2}=189x^2\sqrt{2x}, 4\sqrt[4]{4x^5}\cdot-8\sqrt[4]{3x^3}=-x^2\sqrt[4]{8}$$ $$-6\sqrt{42x^3}\cdot-8\sqrt{28x^2}=189x^2\sqrt{2x}$$$$ 4\sqrt[4]{4x^5}\cdot-8\sqrt[4]{3x^3}=-x^2\sqrt[4]{8}$$ $$-6\sqrt{42x^3}\hspace{.25em}\cdot$$$$-8\sqrt{28x^2}$$$$=189x^2\sqrt{2x}$$$$ 4\sqrt[4]{4x^5}\hspace{.25em}\cdot$$$$-8\sqrt[4]{3x^3}$$$$=-x^2\sqrt[4]{8}$$
E
$$-6\sqrt{42x^3}\cdot-8\sqrt{28x^2}=272x\sqrt{6x}, 4\sqrt[4]{4x^5}\cdot-8\sqrt[4]{3x^3}=-64x^2\sqrt[4]{12}$$ $$-6\sqrt{42x^3}\cdot-8\sqrt{28x^2}=272x\sqrt{6x}$$$$ 4\sqrt[4]{4x^5}\cdot-8\sqrt[4]{3x^3}=-64x^2\sqrt[4]{12}$$ $$-6\sqrt{42x^3}\hspace{.25em}\cdot$$$$-8\sqrt{28x^2}$$$$=272x\sqrt{6x}$$$$ 4\sqrt[4]{4x^5}\hspace{.25em}\cdot$$$$-8\sqrt[4]{3x^3}$$$$=-64x^2\sqrt[4]{12}$$
Question 4 of 5: Simplify, assume all variables represent positive real numbers.
Select the Correct Answer: Correct! Not Correct!
Not Correct! A
$$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{2\sqrt{2}}{3},\frac{2\sqrt{12x^2}}{x\sqrt{4}}=3\sqrt{3}$$ $$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{2\sqrt{2}}{3}$$$$\frac{2\sqrt{12x^2}}{x\sqrt{4}}=3\sqrt{3}$$ $$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{2\sqrt{2}}{3}$$$$\frac{2\sqrt{12x^2}}{x\sqrt{4}}=$$$$3\sqrt{3}$$
B
$$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{2}{3},\frac{2\sqrt{12x^2}}{x\sqrt{4}}=2\sqrt{3}$$ $$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{2}{3}$$$$\frac{2\sqrt{12x^2}}{x\sqrt{4}}=2\sqrt{3}$$ $$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{2}{3}$$$$\frac{2\sqrt{12x^2}}{x\sqrt{4}}=$$$$2\sqrt{3}$$
C
$$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{2\sqrt{5}}{3},\frac{2\sqrt{12x^2}}{x\sqrt{4}}=6\sqrt{2}$$ $$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{2\sqrt{5}}{3}$$$$\frac{2\sqrt{12x^2}}{x\sqrt{4}}=6\sqrt{2}$$ $$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{2\sqrt{5}}{3}$$$$\frac{2\sqrt{12x^2}}{x\sqrt{4}}=$$$$6\sqrt{2}$$
D
$$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{3\sqrt{5}}{9},\frac{2\sqrt{12x^2}}{x\sqrt{4}}=\frac{2\sqrt{3}}{3}$$ $$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{3\sqrt{5}}{9}$$$$\frac{2\sqrt{12x^2}}{x\sqrt{4}}=\frac{2\sqrt{3}}{3}$$ $$\frac{\sqrt{20}}{3\sqrt{5}}=\frac{3\sqrt{5}}{9}$$$$\frac{2\sqrt{12x^2}}{x\sqrt{4}}=$$$$\frac{2\sqrt{3}}{3}$$
E
$$\frac{\sqrt{20}}{3\sqrt{5}}=2\sqrt{2},\frac{2\sqrt{12x^2}}{x\sqrt{4}}=\frac{3\sqrt{3}}{4}$$ $$\frac{\sqrt{20}}{3\sqrt{5}}=2\sqrt{2}$$$$\frac{2\sqrt{12x^2}}{x\sqrt{4}}=\frac{3\sqrt{3}}{4}$$ $$\frac{\sqrt{20}}{3\sqrt{5}}=2\sqrt{2}$$$$\frac{2\sqrt{12x^2}}{x\sqrt{4}}=$$$$\frac{3\sqrt{3}}{4}$$
Question 5 of 5: Simplify, assume all variables represent positive real numbers.
Select the Correct Answer: Correct! Not Correct!
Not Correct! A
$$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}=21yz\sqrt[3]{4xy^2z^2}, \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}=5\sqrt[5]{4a^2b}$$ $$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}=21yz\sqrt[3]{4xy^2z^2}$$$$ \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}=5\sqrt[5]{4a^2b}$$ $$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}$$$$=$$$$21yz\sqrt[3]{4xy^2z^2}$$$$ \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}$$$$=5\sqrt[5]{4a^2b}$$
B
$$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}=14y^2z^2\sqrt[3]{2xyz}, \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}=\frac{2a\sqrt[5]{ab}}{5}$$ $$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}=14y^2z^2\sqrt[3]{2xyz}$$$$ \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}=\frac{2a\sqrt[5]{ab}}{5}$$ $$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}$$$$=$$$$14y^2z^2\sqrt[3]{2xyz}$$$$ \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}$$$$=\frac{2a\sqrt[5]{ab}}{5}$$
C
$$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}=7yz\sqrt[3]{4xy^2z^2}, \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}=\frac{\sqrt[5]{4a^2b}}{5}$$ $$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}=7yz\sqrt[3]{4xy^2z^2}$$$$ \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}=\frac{\sqrt[5]{4a^2b}}{5}$$ $$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}$$$$=7yz\sqrt[3]{4xy^2z^2}$$$$ \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}$$$$=\frac{\sqrt[5]{4a^2b}}{5}$$
D
$$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}=y^2z^2\sqrt[3]{10z^2}, \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}=\frac{2a\sqrt[5]{16ab}}{10}$$ $$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}=y^2z^2\sqrt[3]{10z^2}$$$$ \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}=\frac{2a\sqrt[5]{16ab}}{10}$$ $$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}$$$$=y^2z^2\sqrt[3]{10z^2}$$$$ \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}$$$$=\frac{2a\sqrt[5]{16ab}}{10}$$
E
$$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}=x^2y^2z^2\sqrt[3]{4}, \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}=\frac{a\sqrt[5]{10ab}}{10}$$ $$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}=x^2y^2z^2\sqrt[3]{4}$$$$ \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}=\frac{a\sqrt[5]{10ab}}{10}$$ $$7\sqrt[3]{\frac{24x^4y^5z^6}{6x^3z}}$$$$=x^2y^2z^2\sqrt[3]{4}$$$$ \frac{\sqrt[5]{4a^4b^4}}{\sqrt[5]{3125a^2b^3}}$$$$=\frac{a\sqrt[5]{10ab}}{10}$$
