Sure, let's solve the problem of simplifying the expression [tex]\( 4 \sqrt[3]{192 x^3} + \sqrt[3]{81 x^3} \)[/tex].
1. Break down each term separately:
- For the first term, [tex]\( 4 \sqrt[3]{192 x^3} \)[/tex]:
- Observe that [tex]\( 192 = 2^6 \times 3 \)[/tex], hence we can rewrite [tex]\( 192 x^3 \)[/tex] as [tex]\( 2^6 \times 3 \times x^3 \)[/tex].
- Therefore, [tex]\( \sqrt[3]{192 x^3} \)[/tex] becomes [tex]\( \sqrt[3]{2^6 \times 3 \times x^3} \)[/tex].
Since [tex]\( \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \)[/tex]:
- [tex]\( \sqrt[3]{2^6 \times 3 \times x^3} = \sqrt[3]{2^6} \times \sqrt[3]{3} \times \sqrt[3]{x^3} \)[/tex].
- [tex]\( \sqrt[3]{2^6} = 2^{6/3} = 2^2 = 4 \)[/tex].
- [tex]\( \sqrt[3]{x^3} = x \)[/tex].
Thus,
[tex]\[
\sqrt[3]{192 x^3} = 4 \times \sqrt[3]{3} \times x.
\][/tex]
Incorporating the factor of 4 in front, the term becomes:
[tex]\[
4 \sqrt[3]{192 x^3} = 4 \times 4 \sqrt[3]{3} \times x = 16 \sqrt[3]{3} x.
\][/tex]
- For the second term, [tex]\( \sqrt[3]{81 x^3} \)[/tex]:
- Observe that [tex]\( 81 = 3^4 \)[/tex], hence we can rewrite [tex]\( 81 x^3 \)[/tex] as [tex]\( 3^4 \times x^3 \)[/tex].
- Therefore, [tex]\( \sqrt[3]{81 x^3} \)[/tex] becomes [tex]\( \sqrt[3]{3^4 \times x^3} \)[/tex].
- [tex]\( \sqrt[3]{3^4} = 3^{4/3} = 3 \times \sqrt[3]{3} \)[/tex].
- [tex]\( \sqrt[3]{x^3} = x \)[/tex].
Thus,
[tex]\[
\sqrt[3]{81 x^3} = 3 \sqrt[3]{3} x.
\][/tex]
2. Combine the simplified terms:
- The expression now is:
[tex]\[
4 \sqrt[3]{192 x^3} + \sqrt[3]{81 x^3} = 16 \sqrt[3]{3} x + 3 \sqrt[3]{3} x
\][/tex]
- Combine like terms:
[tex]\[
= (16 + 3) \sqrt[3]{3} x
\][/tex]
[tex]\[
= 19 \sqrt[3]{3} x
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\boxed{19 \sqrt[3]{3} x}
\][/tex]
