{"messages": [{"content": "x^5 + 10x^3 + 20x - 4 = 0. Find a closed form expression for the real roots of x in the simplest form.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2^{\\frac{3}{5}} - 2^{\\frac{2}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 5x^3 + 5x - 2 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(1 + \\sqrt{2})^{\\frac{1}{5}} + -1(1 + \\sqrt{2})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 10x^3 + 20x + 28 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(-14 + 2\\sqrt{57})^{\\frac{1}{5}} + -2(-14 + 2\\sqrt{57})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 15x^3 + 45x - 26 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(13 + 2\\sqrt{103})^{\\frac{1}{5}} + -3(13 + 2\\sqrt{103})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 20x^3 + 80x - 66 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(33 + \\sqrt{2113})^{\\frac{1}{5}} + -4(33 + \\sqrt{2113})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 30x^3 + 180x - 122 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(61 + \\sqrt{11497})^{\\frac{1}{5}} + -6(61 + \\sqrt{11497})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 - 10x^3 + 20x + 36 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(-18 + 2\\sqrt{73})^{\\frac{1}{5}} + 2(-18 + 2\\sqrt{73})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 5x^3 + 5x - 30 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(15 + \\sqrt{226})^{\\frac{1}{5}} + -1(15 + \\sqrt{226})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 40x^3 + 320x - 258 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(129 + \\sqrt{49409})^{\\frac{1}{5}} + -8(129 + \\sqrt{49409})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 15x^3 + 45x + 28 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(-14 + \\sqrt{439})^{\\frac{1}{5}} + -3(-14 + \\sqrt{439})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 - 5x^3 + 5x - 6 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(3 + 2\\sqrt{2})^{\\frac{1}{5}} + -1(3 + 2\\sqrt{2})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 25x^3 + 125x - 126 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(63 + \\sqrt{7094})^{\\frac{1}{5}} + -5(63 + \\sqrt{7094})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 10x^3 + 20x + 56 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(-28 + 4\\sqrt{51})^{\\frac{1}{5}} + -2(-28 + 4\\sqrt{51})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 5x^3 + 5x + 2 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(-1 + \\sqrt{2})^{\\frac{1}{5}} + -1(-1 + \\sqrt{2})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 30x^3 + 180x - 344 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(172 + 4\\sqrt{2335})^{\\frac{1}{5}} + -6(172 + 4\\sqrt{2335})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 - 20x^3 + 80x + 66 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(-33 + \\sqrt{65})^{\\frac{1}{5}} + -4(-33 + \\sqrt{65})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 5x^3 + 5x - 122 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(61 + \\sqrt{3722})^{\\frac{1}{5}} + -1(61 + \\sqrt{3722})^{-\\frac{1}{5}}", "dataset": "MATH"}
{"messages": [{"content": "x^5 + 50x^3 + 500x - 1252 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "(626 + 2\\sqrt{122969})^{\\frac{1}{5}} + -10(626 + 2\\sqrt{122969})^{-\\frac{1}{5}}", "dataset": "MATH"}