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11 problems match “cube roots”. Clear

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1. Edge length of a cube from its volume ●○○○○ 10 ptsThe volume of a cube is $3\frac{3}{8}$ cubic inches. Find the length of each edge of the cube in inches. Give your answer as a decimal.
2. Side of a cube with volume 112 ●○○○○ 10 ptsThe volume of a cube is 112 cubic units. Find the length of each side. Round to two decimal places.
3. Integral of an exponential over a triangle ●○○○○ 10 ptsLet $\gamma$ be the triangle in $\mathbb C$ whose vertices are the three cube roots of $1+i$, oriented clockwise. Find the value of $$\int_\gamma e^{2z+i}\, dz.$$
4. Cube root plus square root equation ●●○○○ 20 ptsSolve the equation $\sqrt[3]{x} + \sqrt{x+12} = 0$ for $x$.
5. Equation with square and cube roots ●●○○○ 20 ptsSolve the equation $\frac{\sqrt{x}+\sqrt[3]{x}}{\sqrt{x}-\sqrt[3]{x}}=3$ for real $x$.
6. Fractional-power equation with substitution ●●○○○ 20 ptsSolve the equation $(x - 2)^{2/3} + (x - 2)^{1/3} - 2 = 0$, using real cube roots.
7. Cube root equation for x to the sixth ●●●○○ 50 ptsA positive real number $x$ satisfies $\sqrt[3]{1-x^3} + \sqrt[3]{1+x^3} = 1$. Find $x^6$. Round to two decimal places.
8. Radical equation with cube roots ●●●○○ 50 ptsSolve the equation $\sqrt[3]{9-\sqrt{x+1}}+\sqrt[3]{7+\sqrt{x+1}}=4$.
9. Equation with cube roots ●●●○○ 50 ptsSolve the equation $\sqrt[3]{2x+1}+\sqrt[3]{4-x}=-\sqrt[3]{3}$ for $x$.
10. Largest finite count of small elements ●●●●○ 100 pts$S$ is a set of complex numbers such that if $u, v \in S$, then $uv \in S$ and $u^{2}+v^{2} \in S$. Suppose that the number $N$ of elements of $S$ with absolute value at most $1$ is finite. What is the largest possible value of $N$?

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