To solve the given problem, we need to simplify and subtract the expressions involving radicals. The expressions are:
[tex]\[
\sqrt[3]{16 x^4} - \sqrt[3]{128 x}
\][/tex]
We will simplify each term individually before performing the subtraction.
### Step-by-Step Solution:
#### Simplifying [tex]\(\sqrt[3]{16 x^4}\)[/tex]
First, we write 16 as a product of its prime factors:
[tex]\[ 16 = 2^4 \][/tex]
So, we rewrite the expression under the cubic root:
[tex]\[ \sqrt[3]{16 x^4} = \sqrt[3]{2^4 \cdot x^4} \][/tex]
We can separate and simplify the terms under the cube root:
[tex]\[ \sqrt[3]{2^4 \cdot x^4} = \sqrt[3]{2^4} \cdot \sqrt[3]{x^4} \][/tex]
Since [tex]\(\sqrt[3]{2^4} = 2^{4/3}\)[/tex] and [tex]\(\sqrt[3]{x^4} = x^{4/3}\)[/tex], the expression becomes:
[tex]\[ \sqrt[3]{16 x^4} = 2^{4/3} x^{4/3} \][/tex]
#### Simplifying [tex]\(\sqrt[3]{128 x}\)[/tex]
Next, we write 128 as a product of its prime factors:
[tex]\[ 128 = 2^7 \][/tex]
So, we rewrite the expression under the cubic root:
[tex]\[ \sqrt[3]{128 x} = \sqrt[3]{2^7 \cdot x} \][/tex]
We can separate and simplify the terms under the cube root:
[tex]\[ \sqrt[3]{2^7 \cdot x} = \sqrt[3]{2^7} \cdot \sqrt[3]{x} \][/tex]
Since [tex]\(\sqrt[3]{2^7} = 2^{7/3}\)[/tex] and [tex]\(\sqrt[3]{x} = x^{1/3}\)[/tex], the expression becomes:
[tex]\[ \sqrt[3]{128 x} = 2^{7/3} x^{1/3} \][/tex]
### Subtracting the Like Radicals
We now subtract the simplified terms:
[tex]\[ 2^{4/3} x^{4/3} - 2^{7/3} x^{1/3} \][/tex]
Since both terms involve [tex]\(2\)[/tex] raised to different powers and [tex]\(x\)[/tex] raised to different powers, they are not like radicals and cannot be directly subtracted. Thus, this is the final simplified form of the expression:
[tex]\[
\boxed{ 2^{4/3} x^{4/3} - 2^{7/3} x^{1/3} }
\][/tex]
This is the exact answer, using radicals as needed.
