To multiply the expressions [tex]\( x^{\frac{2}{5}} \)[/tex] and [tex]\( x^{\frac{2}{9}} \)[/tex], you use the property of exponents that states:
[tex]\[ x^a \cdot x^b = x^{a+b} \][/tex]
1. Identify the exponents:
- The exponent of the first term is [tex]\(\frac{2}{5}\)[/tex].
- The exponent of the second term is [tex]\(\frac{2}{9}\)[/tex].
2. Add the exponents together:
[tex]\[
\frac{2}{5} + \frac{2}{9}
\][/tex]
3. To add these fractions, find a common denominator. The denominators are 5 and 9. The least common multiple of 5 and 9 is 45.
4. Convert each fraction to an equivalent fraction with a denominator of 45:
[tex]\[
\frac{2}{5} = \frac{2 \cdot 9}{5 \cdot 9} = \frac{18}{45}
\][/tex]
[tex]\[
\frac{2}{9} = \frac{2 \cdot 5}{9 \cdot 5} = \frac{10}{45}
\][/tex]
5. Now add the fractions:
[tex]\[
\frac{18}{45} + \frac{10}{45} = \frac{28}{45}
\][/tex]
6. Finally, write the result in exponential form:
[tex]\[
x^{\frac{2}{5}} \cdot x^{\frac{2}{9}} = x^{\frac{28}{45}}
\][/tex]
Thus, the correct answer is:
[tex]\[ x^{\frac{28}{45}} \][/tex]
