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Problem

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problem

The equation x10+(13x−1)10=0x^{10}+(13x-1)^{10}=0\,has 10 complex roots r1,r_1, r¯1,\overline{r}_1, r2,r_2, r¯2,\overline{r}_2, r3,r_3, r¯3,\overline{r}_3, r4,r_4, r¯4,\overline{r}_4, r5,r_5, r¯5,\overline{r}_5, where the bar denotes complex conjugation. Find the value of 1r1r¯1+1r2r¯2+1r3r¯3+1r4r¯4+1r5r¯5.\frac 1{r_1\overline{r}_1}+\frac 1{r_2\overline{r}_2}+\frac 1{r_3\overline{r}_3}+\frac 1{r_4\overline{r}_4}+\frac 1{r_5\overline{r}_5}.
Plain-text mathematical notation (without MathML)
The equation
x¹⁰+(13x−1)¹⁰=0 has 10 complex roots r₁, (r)¯₁, r₂, (r)¯₂, r₃, (r)¯₃, r₄, (r)¯₄, r₅, (r)¯₅, where the bar denotes complex conjugation. Find the value of
(1)/(r₁(r)¯₁)+(1)/(r₂(r)¯₂)+(1)/(r₃(r)¯₃)+(1)/(r₄(r)¯₄)+(1)/(r₅(r)¯₅).
Original LaTeX notation
The equation
\[x^{10}+(13x-1)^{10}=0\,\]has 10 complex roots $r_1,$ $\overline{r}_1,$ $r_2,$ $\overline{r}_2,$ $r_3,$ $\overline{r}_3,$ $r_4,$ $\overline{r}_4,$ $r_5,$ $\overline{r}_5,$ where the bar denotes complex conjugation. Find the value of
\[\frac 1{r_1\overline{r}_1}+\frac 1{r_2\overline{r}_2}+\frac 1{r_3\overline{r}_3}+\frac 1{r_4\overline{r}_4}+\frac 1{r_5\overline{r}_5}.\]

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
a0fd2eb9-9063-5aae-8cb2-3f4c1b5f4672
task_key
intermediate~5falgebra--test--a0fd2eb9-9063-5aae-8cb2-3f4c1b5f4672
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import