Bot GPT 4-o mini To simplify \( 3^{\sqrt{-72}} \), we first need to handle the expression under the square root.
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Simplify \(-72\): \[ -72 = -1 \times 72 = -1 \times (36 \times 2) = -1 \times 36 \times 2 \] The term \(-1\) indicates that we will deal with an imaginary number when we take the square root.
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Take the square root: \[ \sqrt{-72} = \sqrt{-1 \times 72} = \sqrt{-1} \times \sqrt{72} = i \sqrt{72} \]
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Further simplify \(\sqrt{72}\): \[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \] So, we can rewrite \(\sqrt{-72}\): \[ \sqrt{-72} = i \times 6\sqrt{2} = 6i\sqrt{2} \]
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Substituting back into the original expression: \[ 3^{\sqrt{-72}} = 3^{6i\sqrt{2}} \]
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Using the property of exponents: Using the formula \( a^{bi} = e^{b \ln(a)} \): \[ 3^{6i\sqrt{2}} = e^{6i\sqrt{2} \ln(3)} \]
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Final expression: Therefore, the expression \( 3^{\sqrt{-72}} \) can be expressed in exponential form: \[ 3^{\sqrt{-72}} = e^{6i\sqrt{2} \ln(3)} \]
So, the simplified form of \( 3^{\sqrt{-72}} \) is: \[ 3^{\sqrt{-72}} = e^{6i\sqrt{2} \ln(3)} \]
If you prefer a simpler form without using exponents, you may leave the answer as \( 3^{6i\sqrt{2}} \).
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