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8 problems match “exponent laws”. Clear

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1. Middle term of an arithmetic sequence ●○○○○ 10 ptsIn the arithmetic sequence $3^2, x, 3^4$, find the value of $x$.
2. Common ratio from two terms ●○○○○ 10 ptsIn the geometric sequence $\{a_n\}$, $a_3=2$ and $a_6=\frac{1}{4}$. Find the common ratio of the sequence. Give your answer as a decimal.
3. Equation with square and cube roots ●●○○○ 20 ptsSolve the equation $\frac{\sqrt{x}+\sqrt[3]{x}}{\sqrt{x}-\sqrt[3]{x}}=3$ for real $x$.
4. Per-pass speed loss in aerobraking ●●○○○ 20 ptsA spacecraft approaches a planet at 18000 m/s and must slow to 8000 m/s over exactly 50 orbital passes. On each pass it keeps the same fraction of its speed. What percentage of its speed must it lose on each pass? Round to two decimal places.
5. Power of a prime with prime shift ●●○○○ 20 ptsLet $p$ be a prime number such that $p + 3$ is also prime. Find $p^{11} - 52$.
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. Solving a quadratic with exponents ●●○○○ 20 ptsSolve $4(3 - x) + 9^2 = x(x - 4) + 7^2 - 5$ for the positive value of $x$.
8. Third term of a power-plus-constant sequence ●●○○○ 20 ptsLet $\{a_n\}$ be the sequence given by $a_n = a^n + m$ for positive integers $n$, where $a < 0$ and $m$ are constants. Given $a_1 = 2$ and $a_2 = 4$, find $a_3$.