Question

What is 3^√-72 simplified? Note: You will find the option to add a cubic root symbol in the Algebra (×) keyboard.

Answer 1

To simplify \( 3^{\sqrt{-72}} \), we first need to handle the expression under the square root.

1. **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.

2. **Take the square root**:
\[
\sqrt{-72} = \sqrt{-1 \times 72} = \sqrt{-1} \times \sqrt{72} = i \sqrt{72}
\]

3. **Further simplify \(\sqrt{72}\)**:
\[
\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}
\]
So, we can rewrite \(\sqrt{-72}\):
\[
\sqrt{-72} = i \times 6\sqrt{2} = 6i\sqrt{2}
\]

4. **Substituting back into the original expression**:
\[
3^{\sqrt{-72}} = 3^{6i\sqrt{2}}
\]

5. **Using the property of exponents**:
Using the formula \( a^{bi} = e^{b \ln(a)} \):
\[
3^{6i\sqrt{2}} = e^{6i\sqrt{2} \ln(3)}
\]

6. **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 \( 3^{6i\sqrt{2}} \).