a8-47a4+1= a42-2a4×472+4722 +1-4722 = a4-4722 +1-22094= a4-4722-22054= a4-4722-220542= a4-472+22052a4-472-22052 {using the identity a2-b2 = a-ba+b}

evaluate

a⁸ -47a⁴ +1

a^{8} - 47 a^{4} + 1 = {(a^{4})}^{2} - 2 a^{4} \times \frac{47}{2} + {(\frac{47}{2})}^{2} + 1 - {(\frac{47}{2})}^{2} = {(a^{4} - \frac{47}{2})}^{2} + 1 - (\frac{2209}{4}) = {(a^{4} - \frac{47}{2})}^{2} - \frac{2205}{4} = {(a^{4} - \frac{47}{2})}^{2} - {(\sqrt{\frac{2205}{4}})}^{2} = (a^{4} - \frac{47}{2} + \frac{\sqrt{2205}}{2}) (a^{4} - \frac{47}{2} - \frac{\sqrt{2205}}{2}) {using the identity a^{2} - b^{2} = (a - b) (a + b)}

evaluate a8 -47a4 +1