5 answers
Let's break this expression down step by step. You wrote it as √(-8)^2/3, which can be interpreted in several ways. I will assume you meant \(\frac{\sqrt{(-8)^2}}{3}\).
Calculate \((-8)^2\): \[ (-8)^2 = 64 \]
Take the square root of 64: \[ \sqrt{64} = 8 \]
Divide by 3: \[ \frac{8}{3} \]
So, \(\frac{\sqrt{(-8)^2}}{3} = \frac{8}{3}\).
If you meant something different, please clarify!
Thank you for your clarification! If the expression is written as \(\sqrt{-8}^{2/3}\), then we can interpret it as follows:
Step 1: Find the cube root: \[ (-8)^{1/3} = -2 \]
Step 2: Square the result: \[ (-2)^2 = 4 \]
So, \((-8)^{2/3} = 4\).
Hence, \(\sqrt{-8}^{2/3} = 4\).
Thank you for the clarification! If the expression is \(\sqrt{(-8)}^{2/3}\), then we should break it down as follows:
Calculate \(\sqrt{-8}\): Since \(\sqrt{-8}\) involves taking the square root of a negative number, we can rewrite it using imaginary numbers: \[ \sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = \sqrt{8}i = 2\sqrt{2}i \]
Raise \(\sqrt{-8}\) to the power of \(\frac{2}{3}\): \[ \left(2\sqrt{2}i\right)^{\frac{2}{3}} \]
To evaluate this, let's separate the components:
First, raise the modulus to the power of \(\frac{2}{3}\): \[ (2\sqrt{2})^{\frac{2}{3}} = 2^{\frac{2}{3}} \cdot (\sqrt{2})^{\frac{2}{3}} = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{3}{3}} = 2 \]
Now, raise \(i\) to the power of \(\frac{2}{3}\): \[ i^{\frac{2}{3}} = (i^2)^{\frac{1}{3}} = (-1)^{\frac{1}{3}} = -1 \] However, for complex numbers, we typically account for the principal value and potential roots, which would become more complicated.
Putting it together: \[ \sqrt{(-8)}^{\frac{2}{3}} = 2 \cdot (-1) = -2 \]
\[ \sqrt{(-8)}^{\frac{2}{3}} = 2i^{\frac{2}{3}} = 2\sqrt{2}i^{2/3} \] If you need clearly stated values, considering the principal root approach, the simplest real value can yield \(2\). However, please specify if you need consideration of complex interpretations further!
Again, if the intended form not covered, do let me know!
