find square root of (2a+1)(2a+3)(2a+5)(2a+7)+16

asked by guest
on Nov 15, 2024 at 6:11 am



You asked:

Evaluate the expression: \(\sqrt{\left(2 a + 1\right) \cdot \left(2 a + 3\right) \cdot \left(2 a + 5\right) \cdot \left(2 a + 7\right) + 16}\)

MathBot Answer:

Evaluated



\(\displaystyle \sqrt{\left(2 a + 1\right) \cdot \left(2 a + 3\right) \cdot \left(2 a + 5\right) \cdot \left(2 a + 7\right) + 16} = \sqrt{\left(2 a + 1\right) \left(2 a + 3\right) \left(2 a + 5\right) \left(2 a + 7\right) + 16} \)

Expanded

\[\sqrt{\left(2 a + 1\right) \cdot \left(2 a + 3\right) \cdot \left(2 a + 5\right) \cdot \left(2 a + 7\right) + 16} = \sqrt{16 a^{4} + 128 a^{3} + 344 a^{2} + 352 a + 121}\]

Factored

\[\sqrt{\left(2 a + 1\right) \cdot \left(2 a + 3\right) \cdot \left(2 a + 5\right) \cdot \left(2 a + 7\right) + 16} = \sqrt{\left(4 a^{2} + 16 a + 11\right)^{2}}\]

Roots

\[\sqrt{\left(2 a + 1\right) \cdot \left(2 a + 3\right) \cdot \left(2 a + 5\right) \cdot \left(2 a + 7\right) + 16} = i \sqrt[4]{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)}\right)^{2} + \left(16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)},16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121 \right)}}{2} \right)} + \sqrt[4]{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)}\right)^{2} + \left(16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)},16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121 \right)}}{2} \right)} \approx 26.229754097208 i \left(\left(0.0930232558139535 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 0.558139534883721 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 0.0930232558139535 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 0.186046511627907 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 0.511627906976744 \operatorname{im}{\left(a\right)}\right)^{2} + 0.311519740400216 \left(0.0416666666666667 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 0.333333333333333 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 0.25 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.895833333333333 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.916666666666667 \operatorname{re}{\left(a\right)} + 0.0416666666666667 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 0.895833333333333 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.315104166666667\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)},16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121 \right)}}{2} \right)} + 26.229754097208 \left(\left(0.0930232558139535 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 0.558139534883721 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 0.0930232558139535 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 0.186046511627907 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 0.511627906976744 \operatorname{im}{\left(a\right)}\right)^{2} + 0.311519740400216 \left(0.0416666666666667 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 0.333333333333333 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 0.25 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.895833333333333 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.916666666666667 \operatorname{re}{\left(a\right)} + 0.0416666666666667 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 0.895833333333333 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.315104166666667\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)},16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121 \right)}}{2} \right)}\]\[\sqrt{\left(2 a + 1\right) \cdot \left(2 a + 3\right) \cdot \left(2 a + 5\right) \cdot \left(2 a + 7\right) + 16} = i \left(- \sqrt[4]{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)}\right)^{2} + \left(16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)},16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121 \right)}}{2} \right)}\right) - \sqrt[4]{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)}\right)^{2} + \left(16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)},16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121 \right)}}{2} \right)} \approx - 26.229754097208 i \left(\left(0.0930232558139535 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 0.558139534883721 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 0.0930232558139535 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 0.186046511627907 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 0.511627906976744 \operatorname{im}{\left(a\right)}\right)^{2} + 0.311519740400216 \left(0.0416666666666667 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 0.333333333333333 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 0.25 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.895833333333333 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.916666666666667 \operatorname{re}{\left(a\right)} + 0.0416666666666667 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 0.895833333333333 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.315104166666667\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)},16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121 \right)}}{2} \right)} - 26.229754097208 \left(\left(0.0930232558139535 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 0.558139534883721 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 0.0930232558139535 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 0.186046511627907 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 0.511627906976744 \operatorname{im}{\left(a\right)}\right)^{2} + 0.311519740400216 \left(0.0416666666666667 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 0.333333333333333 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 0.25 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.895833333333333 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.916666666666667 \operatorname{re}{\left(a\right)} + 0.0416666666666667 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 0.895833333333333 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.315104166666667\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(64 \left(\operatorname{re}{\left(a\right)}\right)^{3} \operatorname{im}{\left(a\right)} + 384 \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a\right)} - 64 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{3} + 688 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 128 \left(\operatorname{im}{\left(a\right)}\right)^{3} + 352 \operatorname{im}{\left(a\right)},16 \left(\operatorname{re}{\left(a\right)}\right)^{4} + 128 \left(\operatorname{re}{\left(a\right)}\right)^{3} - 96 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 344 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 384 \operatorname{re}{\left(a\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} + 352 \operatorname{re}{\left(a\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{4} - 344 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 121 \right)}}{2} \right)}\]

+ Submit Question