Counting Rational Terms in the Expansion of (sqrt(2) * 3^(1/3))^100

Consider the expansion of the expression (sqrt(2) * 3^(1/3))^100. This problem explores how to determine the number of rational terms in this expansion. We will use expansion techniques and properties of exponents to solve this problem.

Understanding the Expansion

The given expression is:

(sqrt(2) * 3^(1/3))^100

To expand this, we can use the binomial theorem, which states:

(a b)^n sum_{k0}^{n} binom{n}{k} a^{n-k} b^k

Here, (a sqrt(2)) and (b 3^(1/3)), and (n 100). So, the expansion can be written as:

(sqrt(2) * 3^(1/3))^100 sum_{k0}^{100} binom{100}{k} (sqrt(2))^{100-k} (3^(1/3))^k

Further simplifying the exponents, we get:

(sqrt(2) * 3^(1/3))^100 sum_{k0}^{100} binom{100}{k} 2^{(100-k)/2} 3^{k/3}

Identifying Rational Terms

A term in the expansion is rational if both exponents, (frac{100-k}{2}) and (frac{k}{3}), are integers. This means (100-k) must be even, and (k) must be a multiple of 3.

Case 1: Order of Operations Without Parentheses

When the order of operations is interpreted without parentheses, the expression is:

(sqrt(2) * 3^(1/3))^100 (sqrt(2) * 1)^100 (sqrt(2))^100

The term (sqrt(2)) is rational only when its power is even. Therefore, the relevant even powers are {0, 2, 4, …, 96, 98, 100}. This gives us:

Answer: 51 rational terms

Case 2: Order of Operations With Parentheses

When the division of 1 by 3 is within parentheses, the expression is interpreted as:

(sqrt(2) * (3^(1/3)))^100

The expansion is:

(sqrt(2) * (3^(1/3)))^100 (sqrt(2))^100 * (3^(1/3))^100

The term ((sqrt(2))^100) has 51 even powers, and the term ((3^(1/3))^100) is rational for multiples of 3. Therefore, the combined conditions require the power to be a multiple of 6.

The valid powers in the range 0 to 100 that are multiples of 6 are: {0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96}. This gives us:

Answer: 17 rational terms

Conclusion

In summary, the number of rational terms in the expansion of (sqrt(2) * 3^(1/3))^100 changes based on the order of operations. Using the correct interpretation and parentheses, we can accurately count the rational terms, demonstrating the importance of clear notation in mathematics.

Keywords: Rational terms, expansion, mathematical series