To find the sum of [tex]\(4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x}\)[/tex] in simplest form, let's break it down step-by-step and simplify each term separately.
1. Simplify the first term [tex]\(4 \sqrt[3]{x^{10}}\)[/tex]:
[tex]\[ 4 \sqrt[3]{x^{10}} \][/tex]
The cube root of [tex]\(x^{10}\)[/tex] can be written as:
[tex]\[ \sqrt[3]{x^{10}} = (x^{10})^{1/3} = x^{10/3} \][/tex]
So, the first term becomes:
[tex]\[ 4 \sqrt[3]{x^{10}} = 4 \cdot x^{10/3} = 4 x^{10/3} \][/tex]
2. Simplify the second term [tex]\(5 x^3 \sqrt[3]{8 x}\)[/tex]:
[tex]\[ 5 x^3 \sqrt[3]{8 x} \][/tex]
The cube root of [tex]\(8 x\)[/tex] is:
[tex]\[ \sqrt[3]{8 x} \][/tex]
We know that [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex], so:
[tex]\[ \sqrt[3]{8 x} = \sqrt[3]{2^3 \cdot x} = 2 \cdot \sqrt[3]{x} = 2 x^{1/3} \][/tex]
Therefore, the second term becomes:
[tex]\[ 5 x^3 \cdot \sqrt[3]{8 x} = 5 x^3 \cdot 2 x^{1/3} = 10 x^{3 + 1/3} = 10 x^{10/3} \][/tex]
3. Combine the simplified terms:
Now, sum the two simplified terms:
[tex]\[ 4 x^{10/3} + 10 x^{10/3} \][/tex]
These are like terms and can be added together:
[tex]\[ = (4 + 10) x^{10/3} \][/tex]
[tex]\[ = 14 x^{10/3} \][/tex]
Thus, the simplest form of [tex]\(4 \sqrt[3]{x^{10}} + 5 x^3 \sqrt[3]{8 x}\)[/tex] is:
[tex]\[ 14 x^{10/3} \][/tex]
