To simplify the expression [tex]\(3 x \sqrt[3]{2} + x \sqrt[3]{16}\)[/tex], we need to follow several steps.
First, let’s understand the terms individually:
1. The term [tex]\(3 x \sqrt[3]{2}\)[/tex] represents three times [tex]\(x\)[/tex] times the cube root of 2.
2. The term [tex]\(x \sqrt[3]{16}\)[/tex] represents [tex]\(x\)[/tex] times the cube root of 16.
Next, we simplify the cube root of 16. Recall that:
[tex]\[ 16 = 2^4 \][/tex]
Thus:
[tex]\[ \sqrt[3]{16} = \sqrt[3]{2^4} \][/tex]
Using the properties of exponents:
[tex]\[ \sqrt[3]{2^4} = (2^4)^{1/3} = 2^{4/3} \][/tex]
Now we can rewrite the term [tex]\(x \sqrt[3]{16}\)[/tex]:
[tex]\[ x \sqrt[3]{16} = x 2^{4/3} \][/tex]
So the original expression now looks like this:
[tex]\[ 3 x \sqrt[3]{2} + x 2^{4/3} \][/tex]
Next, let’s express everything using a common base. Notice that [tex]\(2^{4/3}\)[/tex] is the same as [tex]\(2^{1/3} \cdot 2\)[/tex]:
[tex]\[ 2^{4/3} = 2^{1/3} \cdot 2 \][/tex]
Thus:
[tex]\[ x 2^{4/3} = x (2^{1/3} \cdot 2) = 2 x \sqrt[3]{2} \][/tex]
We now rewrite the expression:
[tex]\[ 3 x \sqrt[3]{2} + x 2^{4/3} = 3 x \sqrt[3]{2} + 2 x \sqrt[3]{2} \][/tex]
Since both terms have the common factor [tex]\( x \sqrt[3]{2}\)[/tex], we can factor [tex]\( x \sqrt[3]{2}\)[/tex] out:
[tex]\[ 3 x \sqrt[3]{2} + 2 x \sqrt[3]{2} = (3 + 2) x \sqrt[3]{2} \][/tex]
Which simplifies to:
[tex]\[ 5 x \sqrt[3]{2} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{5 x \sqrt[3]{2}} \][/tex]
So, the simplified form of the expression [tex]\(3 x \sqrt[3]{2} + x \sqrt[3]{16}\)[/tex] is
[tex]\[ \boxed{5 x \sqrt[3]{2}} \][/tex]
and the correct answer option is:
C. [tex]\(5 x \sqrt[3]{2}\)[/tex]
