Asked by Anonymous
Solve the radical equation root5x+5 -x=1
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Answered by
GPT-5 mini
AI
Interpret the equation as 5th root of (x+5): (x+5)^(1/5) − x = 1.
Rearrange:
(x+5)^(1/5) = x + 1.
Raise both sides to the 5th power (odd power, so no extraneous real roots are introduced):
x + 5 = (x + 1)^5
⇔ (x + 1)^5 − (x + 5) = 0.
This quintic has a single real root. Numerically solving (by Newton/bisection) gives
x ≈ 0.40119293.
Check: (x+5)^(1/5) ≈ 1.40119293 and (x+5)^(1/5) − x ≈ 1.40119293 − 0.40119293 = 1.
Solution: x ≈ 0.401193 (rounded).
Rearrange:
(x+5)^(1/5) = x + 1.
Raise both sides to the 5th power (odd power, so no extraneous real roots are introduced):
x + 5 = (x + 1)^5
⇔ (x + 1)^5 − (x + 5) = 0.
This quintic has a single real root. Numerically solving (by Newton/bisection) gives
x ≈ 0.40119293.
Check: (x+5)^(1/5) ≈ 1.40119293 and (x+5)^(1/5) − x ≈ 1.40119293 − 0.40119293 = 1.
Solution: x ≈ 0.401193 (rounded).
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