To simplify the expression [tex]\( 3 \sqrt[3]{3} + 2 \sqrt[3]{81} \)[/tex], let's break it down step-by-step.
1. Evaluate the Terms Separately:
- First, take the term [tex]\( 3 \sqrt[3]{3} \)[/tex]:
[tex]\[
3 \sqrt[3]{3}
\][/tex]
- Second, take the term [tex]\( 2 \sqrt[3]{81} \)[/tex]:
[tex]\[
2 \sqrt[3]{81}
\][/tex]
2. Simplify [tex]\( \sqrt[3]{81} \)[/tex]:
- Notice that [tex]\( 81 \)[/tex] can be expressed as [tex]\( 3^4 \)[/tex]:
[tex]\[
81 = 3^4
\][/tex]
- Therefore, [tex]\( \sqrt[3]{81} \)[/tex] can be written as [tex]\( \sqrt[3]{3^4} \)[/tex]:
[tex]\[
\sqrt[3]{81} = \sqrt[3]{3^4}
\][/tex]
- Using the property of exponents, [tex]\( \sqrt[3]{3^4} = 3^{4/3} \)[/tex]:
[tex]\[
\sqrt[3]{81} = 3^{4/3}
\][/tex]
3. Combine the [tex]\( \sqrt[3]{81} \)[/tex] term:
- Replace [tex]\( \sqrt[3]{81} \)[/tex] in the original term [tex]\( 2 \sqrt[3]{81} \)[/tex] with [tex]\( 3^{4/3} \)[/tex]:
[tex]\[
2 \sqrt[3]{81} = 2 \cdot 3^{4/3}
\][/tex]
4. Express [tex]\( 3^{4/3} \)[/tex] in a more familiar form:
- We know [tex]\( 3^{4/3} \)[/tex] can be broken down as:
[tex]\[
3^{4/3} = 3^{1 + 1/3} = 3 \cdot 3^{1/3}
\][/tex]
5. Simplify [tex]\( 2 \cdot 3^{4/3} \)[/tex]:
- Substitute [tex]\( 3 \cdot 3^{1/3} \)[/tex] back into the term [tex]\( 2 \cdot 3^{4/3} \)[/tex]:
[tex]\[
2 \cdot 3^{4/3} = 2 \cdot (3 \cdot 3^{1/3}) = 2 \cdot 3 \cdot 3^{1/3} = 6 \cdot 3^{1/3}
\][/tex]
6. Combine the Expressions:
- Now combine the simplified terms [tex]\( 3 \sqrt[3]{3} \)[/tex] and [tex]\( 6 \sqrt[3]{3} \)[/tex]:
[tex]\[
3 \sqrt[3]{3} + 6 \sqrt[3]{3}
\][/tex]
7. Add Like Terms:
- Since both terms have [tex]\( \sqrt[3]{3} \)[/tex], add the coefficients:
[tex]\[
3 + 6 = 9
\][/tex]
- Therefore, the combined expression is:
[tex]\[
9 \sqrt[3]{3}
\][/tex]
Final Answer:
So, [tex]\(\boxed{9 \sqrt[3]{3}}\)[/tex].
