Certainly! Let's solve this step by step.
We need to find the value of \(\sqrt[3]{x^{10}}\) for \(x = -2\) and express it in the form \(a \sqrt[3]{b}\).
### Step 1: Calculate \(x^{10}\)
Given:
[tex]\[ x = -2 \][/tex]
First, let's compute \(x^{10}\):
[tex]\[ x^{10} = (-2)^{10} \][/tex]
Since the exponent is even, the result will be positive.
[tex]\[ (-2)^{10} = 1024 \][/tex]
### Step 2: Calculate the cube root of \(x^{10}\)
Next, we need to find \(\sqrt[3]{1024}\):
[tex]\[ \sqrt[3]{1024} \][/tex]
The cube root of 1024 is approximately:
[tex]\[ \sqrt[3]{1024} \approx 10.08 \][/tex]
### Step 3: Expressing \(\sqrt[3]{x^{10}}\) in the form \(a \sqrt[3]{b}\)
We can rewrite \(x^{10}\) as:
[tex]\[ x^{10} = (-2)^{10} = (-2)^9 \cdot (-2) = (-8)^3 \cdot (-2) \][/tex]
So we express \(\sqrt[3]{1024}\) as:
[tex]\[ \sqrt[3]{1024} = \sqrt[3]{(-8)^3 \cdot (-2)} = (-8) \cdot \sqrt[3]{-2} \][/tex]
Therefore, the value of \(\sqrt[3]{x^{10}}\) for \(x = -2\) can be written as:
[tex]\[ \sqrt[3]{x^{10}} = -8 \sqrt[3]{-2} \][/tex]
This gives us \(a = -8\) and \(b = -2\).
So the solution is:
[tex]\[ a = -8 \][/tex]
[tex]\[ b = -2 \][/tex]
Thus, the value of [tex]\(\sqrt[3]{x^{10}}\)[/tex] when [tex]\(x = -2\)[/tex] can be written in the form [tex]\(a \sqrt[3]{b}\)[/tex] where [tex]\(a = -8\)[/tex] and [tex]\(b = -2\)[/tex].
