To determine the simplest form of \(\sqrt[3]{x^{10}}\) when \(x = -2\), let's work through the problem step by step:
1. Start with the given value of \(x\):
[tex]\[
x = -2
\][/tex]
2. Calculate \(x^{10}\):
[tex]\[
x^{10} = (-2)^{10}
\][/tex]
When a negative number is raised to an even power, the result is positive. Therefore:
[tex]\[
(-2)^{10} = 1024
\][/tex]
3. Find the cube root of \(x^{10}\):
[tex]\[
\sqrt[3]{(-2)^{10}} = \sqrt[3]{1024}
\][/tex]
4. Express \(\sqrt[3]{1024}\) in the form \(a \sqrt[3]{b}\):
We need to find an \(a\) and a \(b\) such that:
[tex]\[
\sqrt[3]{1024} = a \sqrt[3]{b}
\][/tex]
Notice that 1024 is a perfect power of 2:
[tex]\[
1024 = 2^{10}
\][/tex]
Now take the cube root of \(2^{10}\):
[tex]\[
\sqrt[3]{1024} = \sqrt[3]{2^{10}} = 2^{10/3}
\][/tex]
To express this as \(a \sqrt[3]{b}\), decompose \(2^{10/3}\):
[tex]\[
2^{10/3} = 2^{3 + 1/3} = 2^3 \cdot 2^{1/3}
\][/tex]
Simplify this:
[tex]\[
2^3 \cdot 2^{1/3} = 8 \cdot \sqrt[3]{2}
\][/tex]
5. Identify \(a\) and \(b\):
By comparing the simplified expression with the form \(a \sqrt[3]{b}\), we see that:
[tex]\[
a = 8 \quad \text{and} \quad b = 2
\][/tex]
Therefore, \(\sqrt[3]{x^{10}}\) when \(x = -2\) can be written in simplest form as:
[tex]\[
8 \sqrt[3]{2}
\][/tex]
So, the values of \(a\) and \(b\) are:
[tex]\[
a = 8 \quad \text{and} \quad b = 2
\][/tex]
