- #1

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- Homework Statement
- Why does the TI-83+ calculator think ## (\sqrt[4] {-1})^4 = -1 + 2\times 10^{-13}i ##?

(Calculator is in mode a+bi and degrees. )

- Relevant Equations
- ##(\sqrt[4] {-1})^4##

I think the solution should be:

METHOD #1:

\begin{align} (\sqrt[4] {-1})^4 & = (-1)^{\frac 4 4} \nonumber \\ & = (-1)^1 \text{, can reduce 4/4 since base is a constant and not a variable in ℝ} \nonumber \\ & = -1 \nonumber \end{align}

Alternatively, METHOD #2 for same answer is:

\begin{align} (\sqrt[4] {-1})^4 & = ((-1)^{\frac 1 4})^4 \nonumber \\ & = \left\{\left[(-1)^{\frac 1 2}\right]^{\frac 1 2}\right\}^4 \nonumber \\ & = (\sqrt[] {i})^4 \nonumber \\ &= i^{\frac 4 2} \nonumber \\ &= i^2 \nonumber \\ &= -1 \nonumber \end{align}

However, TI-83+ calculator thinks that the solution is:

METHOD #1:

\begin{align} (\sqrt[4] {-1})^4 & = (-1)^{\frac 4 4} \nonumber \\ & = (-1)^1 \text{, can reduce 4/4 since base is a constant and not a variable in ℝ} \nonumber \\ & = -1 \nonumber \end{align}

Alternatively, METHOD #2 for same answer is:

\begin{align} (\sqrt[4] {-1})^4 & = ((-1)^{\frac 1 4})^4 \nonumber \\ & = \left\{\left[(-1)^{\frac 1 2}\right]^{\frac 1 2}\right\}^4 \nonumber \\ & = (\sqrt[] {i})^4 \nonumber \\ &= i^{\frac 4 2} \nonumber \\ &= i^2 \nonumber \\ &= -1 \nonumber \end{align}

However, TI-83+ calculator thinks that the solution is:

## (\sqrt[4] {-1})^4 = -1 + 2\times 10^{-13}i ##

The calculator on https://www.symbolab.com/ gets the answer -1 which is the same as what I calculated. So, why is the TI-83+ calculator getting a different answer? (And I'm assuming that my answer is correct?

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