What is factor i^{4k+5}+i^{8k+23}-i^{4k-2} ?

The solution to factor i^{4k+5}+i^{8k+23}-i^{4k-2} is i^{4k+2}(i^3+i^{4k+21}-1)

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factor i^{4k+5}+i^{8k+23}-i^{4k-2}

factor

i^{4 k + 5} + i^{8 k + 23} - i^{4 k - 2}

i^{4 k + 2} (i^{3} + i^{4 k + 21} - 1)

Solution steps

i^{4 k + 5} + i^{8 k + 23} - i^{4 k - 2}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

i^{4 k + 5} = i^{4 k} i^{5}

= i^{4 k} i^{5} + i^{8 k + 23} - i^{4 k - 2}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

i^{8 k + 23} = i^{8 k} i^{23}

= i^{4 k} i^{5} + i^{8 k} i^{23} - i^{4 k - 2}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

i^{4 k - 2} = i^{4 k} i^{- 2}

= i^{4 k} i^{5} + i^{8 k} i^{23} - i^{4 k} i^{- 2}

Factor out common term

i^{4 k} i^{- 2} : i^{4 k} i^{- 2} (i^{7} + i^{4 k} i^{25} - i^{4})

= i^{4 k} i^{- 2} (i^{7} + i^{4 k} i^{25} - i^{4})

Factor

i^{7} + i^{4 k} i^{25} - i^{4} : i^{4} (i^{3} + i^{4 k + 21} - 1)

= i^{4 k} i^{- 2} i^{4} (i^{3} + i^{4 k + 21} - 1)

Simplify

i^{4 k} i^{- 2} i^{4} : i^{4 k + 2}

= i^{4 k + 2} (i^{3} + i^{4 k + 21} - 1)

2 0^{2} + 1 5^{2} = x^{2}

x^{2} + x - 1 \geq 0

9 + n < 10

- 3 x^{2} + 3 = (x + 1) (x - 1)

- 27 = 9 (- 1 + x)

What is factor i^{4k+5}+i^{8k+23}-i^{4k-2} ?
The solution to factor i^{4k+5}+i^{8k+23}-i^{4k-2} is i^{4k+2}(i^3+i^{4k+21}-1)