To solve and simplify the expression [tex]\(\frac{2 x^2}{3} \div \frac{x^3}{9}\)[/tex]:
1. Understand the problem: We need to divide one fraction by another.
2. Reciprocal of the Divisor: Dividing by a fraction is the same as multiplying by its reciprocal. So, instead of dividing by [tex]\(\frac{x^3}{9}\)[/tex], we can multiply by its reciprocal, [tex]\(\frac{9}{x^3}\)[/tex].
[tex]\[
\frac{2 x^2}{3} \div \frac{x^3}{9} = \frac{2 x^2}{3} \times \frac{9}{x^3}
\][/tex]
3. Multiply the Fractions: To multiply two fractions, multiply their numerators and their denominators:
[tex]\[
\frac{2 x^2 \times 9}{3 \times x^3} = \frac{18 x^2}{3 x^3}
\][/tex]
4. Simplify the Fraction:
- Simplify the numerical part: [tex]\(\frac{18}{3} = 6\)[/tex]
- Simplify the variable part: [tex]\(\frac{x^2}{x^3} = \frac{1}{x^{3-2}} = \frac{1}{x}\)[/tex]
So, the simplified fraction is:
[tex]\[
\frac{18 x^2}{3 x^3} = 6 \times \frac{1}{x} = \frac{6}{x}
\][/tex]
Therefore, the final answer is:
[tex]\(\boxed{\frac{6}{x}}\)[/tex]
