1.
Item 1. Simplify. \(9\sqrt[3]{8}-5\left(2+\sqrt[3]{250}\right)\)
Solution: \(9\sqrt[3]{8}-5(2+\sqrt[3]{250})=9(2)-5(2+5\sqrt[3]{2})=18-10-25\sqrt[3]{2}=8-25\sqrt[3]{2}\). Answer: C
2.
Item 8. Solve the following radical equation. \(\sqrt{x-5}-\sqrt{3x+5}=0\)
Solution: Restrictions require \(x\geq5\). Solve: \(\sqrt{x-5}=\sqrt{3x+5}\), so \(x-5=3x+5\), \(-2x=10\), and \(x=-5\). This does not satisfy \(x\geq5\), so there is no solution. Answer: E
3.
Item 4. Simplify and rationalize the denominator. \(\frac{4\sqrt{20}-8\sqrt{72}}{\sqrt{3}}\)
Solution: \(4\sqrt{20}=8\sqrt5\) and \(8\sqrt{72}=48\sqrt2\). Then \(\frac{8\sqrt5-48\sqrt2}{\sqrt3}\cdot\frac{\sqrt3}{\sqrt3}=\frac{8\sqrt{15}-48\sqrt6}{3}\). Answer: D
4.
Item 5. Which of the following is equivalent to \(\frac{1}{\left(\sqrt[6]{x}\right)^5}\)?
Solution: \(\sqrt[6]{x}=x^{1/6}\), so \(\frac{1}{(\sqrt[6]{x})^5}=(x^{1/6})^{-5}=x^{-5/6}\). Answer: A
5.
Item 6. Simplify. \((64a^{12}b^{15})^{\frac{2}{3}}\)
Solution: \((64a^{12}b^{15})^{2/3}=(\sqrt[3]{64})^2a^{12\cdot2/3}b^{15\cdot2/3}=4^2a^8b^{10}=16a^8b^{10}\). Answer: C
6.
Item 2. Simplify. \(-3\left(\sqrt{5v^6}+\sqrt{20v^6}\right)+\sqrt{125v^6}\)
Solution: \(\sqrt{5v^6}=v^3\sqrt5\), \(\sqrt{20v^6}=2v^3\sqrt5\), and \(\sqrt{125v^6}=5v^3\sqrt5\). So \(-3(v^3\sqrt5+2v^3\sqrt5)+5v^3\sqrt5=-4v^3\sqrt5\). Answer: A
7.
Item 7. What are the restrictions on \(x\)? Assume the expression is real. \(7-\sqrt{3x-8}\)
Solution: The radicand must be nonnegative. \(3x-8\geq0\), so \(3x\geq8\), and \(x\geq\frac{8}{3}\). Answer: E
8.
Item 3. Multiply the following expression and put it in simplest mixed radical form. \(-2\sqrt[3]{24}\left(-4\sqrt[3]{27}\right)\)
Solution: \((-2)(-4)\sqrt[3]{24}\sqrt[3]{27}=8\sqrt[3]{648}=8\sqrt[3]{216\cdot3}=8(6)\sqrt[3]{3}=48\sqrt[3]{3}\). Answer: D
1 out of 1
