The number of rational terms in the expansion (2^1/5 + 3^1/10)⁴⁵ is

To find the number of rational terms in the expansion of \( (2 \frac{1}{5} + 3 \frac{1}{10})^{45} \), we first convert the mixed numbers into improper fractions.

Conversion to Improper Fractions

We can express the mixed numbers as follows:

• \( 2 \frac{1}{5} = \frac{11}{5} \)
• \( 3 \frac{1}{10} = \frac{31}{10} \)

Combining the Fractions

Now, we rewrite the expression:

\( \left( \frac{11}{5} + \frac{31}{10} \right)^{45} \)

To add these fractions, we need a common denominator:

\( \frac{11}{5} = \frac{22}{10} \)

Thus, the sum becomes:

\( \frac{22}{10} + \frac{31}{10} = \frac{53}{10} \)

Expansion of the Binomial

We are now looking at:

\( \left( \frac{53}{10} \right)^{45} \)

Finding Rational Terms

In the binomial expansion of \( (a + b)^n \), the general term is given by:

\( T_k = \binom{n}{k} a^{n-k} b^k \)

For our case, we need to determine when the terms are rational. The only variable is the powers of \( \frac{53}{10} \), which will always yield rational results since both \( 53 \) and \( 10 \) are integers.

Conclusion on Rational Terms

Since all terms in the expansion will be rational, the total number of rational terms is equal to the number of terms in the expansion, which is \( n + 1 \). Therefore, for \( n = 45 \):

Number of rational terms = \( 45 + 1 = 46 \).

Final Answer: There are 46 rational terms in the expansion of \( (2 \frac{1}{5} + 3 \frac{1}{10})^{45} \).