Understanding the Problem: Number of Terms in Expansion

The question asks for the total number of distinct terms when the expression \(\rm \left(\frac{a^2}{b^2}+\frac{b^2}{a^2}+2\right)^{21}\) is fully expanded. The given condition \(\rm a \ne 0, b \ne 0\) ensures that the terms \(\frac{a^2}{b^2}\) and \(\frac{b^2}{a^2}\) are well-defined.

Simplifying the Expression

First, let's look at the expression inside the parentheses: \(\rm \frac{a^2}{b^2}+\frac{b^2}{a^2}+2\). We can try to simplify this expression. Notice that it looks similar to the expansion of a perfect square, \((x+y)^2 = x^2 + y^2 + 2xy\).

Let's consider if this expression can be written as a square of a simpler term. Let \(x = \frac{a}{b}\) and \(y = \frac{b}{a}\). Then:

• \(x^2 = \left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}\)
• \(y^2 = \left(\frac{b}{a}\right)^2 = \frac{b^2}{a^2}\)
• \(xy = \left(\frac{a}{b}\right) \left(\frac{b}{a}\right) = 1\)

So, the expression \(\rm \frac{a^2}{b^2}+\frac{b^2}{a^2}+2\) can be written as:

\[ \rm \frac{a^2}{b^2}+\frac{b^2}{a^2}+2 = \left(\frac{a}{b}\right)^2 + \left(\frac{b}{a}\right)^2 + 2 \cdot \left(\frac{a}{b}\right) \cdot \left(\frac{b}{a}\right) = \left(\frac{a}{b} + \frac{b}{a}\right)^2 \]

Applying the Simplification

Now, substitute this simplified form back into the original expression:

\[ \rm \left(\frac{a^2}{b^2}+\frac{b^2}{a^2}+2\right)^{21} = \left( \left(\frac{a}{b} + \frac{b}{a}\right)^2 \right)^{21} \]

Using the rule of exponents \((x^m)^n = x^{mn}\), we simplify the expression further:

\[ \rm \left( \left(\frac{a}{b} + \frac{b}{a}\right)^2 \right)^{21} = \left(\frac{a}{b} + \frac{b}{a}\right)^{2 \times 21} = \left(\frac{a}{b} + \frac{b}{a}\right)^{42} \]

So, the problem reduces to finding the number of terms in the expansion of the binomial \(\rm \left(\frac{a}{b} + \frac{b}{a}\right)^{42}\).

Using the Binomial Theorem for Number of Terms

According to the Binomial Theorem, the expansion of a binomial \((x+y)^n\) has exactly \(n+1\) terms, provided that \(x\) and \(y\) are distinct variables or expressions that do not simplify further to reduce the number of terms.

In our simplified expression, we have \(\rm \left(\frac{a}{b} + \frac{b}{a}\right)^{42}\). Here, \(x = \frac{a}{b}\), \(y = \frac{b}{a}\), and \(n = 42\).

Since \(a \ne 0\) and \(b \ne 0\), \(\frac{a}{b}\) is a variable quantity (or a constant not equal to \(\pm 1\) if \(a\) and \(b\) are related). The terms in the expansion of \(\rm \left(\frac{a}{b} + \frac{b}{a}\right)^{42}\) will be of the form \(\binom{42}{k} \left(\frac{a}{b}\right)^k \left(\frac{b}{a}\right)^{42-k}\) for \(k = 0, 1, \dots, 42\). These terms simplify to \(\binom{42}{k} \left(\frac{a}{b}\right)^k \left(\frac{a}{b}\right)^{-(42-k)} = \binom{42}{k} \left(\frac{a}{b}\right)^{2k-42}\). Since \(k\) takes values from 0 to 42, the exponent \(2k-42\) takes distinct even integer values from \(2(0)-42 = -42\) up to \(2(42)-42 = 42\). Thus, there are no like terms to combine, and all \(n+1\) terms are distinct.

The number of terms in the expansion of \(\rm \left(\frac{a}{b} + \frac{b}{a}\right)^{42}\) is \(n+1\), where \(n=42\).

Number of terms \(= 42 + 1 = 43\).

Conclusion

The expansion of \(\rm \left(\frac{a^2}{b^2}+\frac{b^2}{a^2}+2\right)^{21}\) simplifies to the expansion of \(\rm \left(\frac{a}{b} + \frac{b}{a}\right)^{42}\). A binomial raised to the power of \(n\) has \(n+1\) terms. Since the power is 42, the number of terms is \(42+1=43\).

The final answer is 43.

Revision Table: Key Concepts

Additional Information: Binomial and Trinomial Expansions

The number of terms in an algebraic expansion depends on the number of terms in the base and the power it is raised to.

• Binomial Expansion: For \((x+y)^n\), the number of terms is \(n+1\). The terms are of the form \(\binom{n}{k} x^{n-k} y^k\), and for generic \(x, y\), all these terms are distinct.
• Trinomial Expansion: For \((x+y+z)^n\), the number of terms is given by the formula \(\frac{(n+2)(n+1)}{2}\). The general term is \(\frac{n!}{i!j!k!} x^i y^j z^k\), where \(i+j+k=n\).
• General Multinomial Expansion: For \((x_1 + x_2 + \dots + x_m)^n\), the number of terms is \(\binom{n+m-1}{m-1}\).

In this problem, although the original expression is a trinomial raised to a power, the base simplifies to a binomial squared. This simplification is crucial. If the base did not simplify, we would use the trinomial expansion formula. For instance, the number of terms in \((a+b+c)^{21}\) would be \(\frac{(21+2)(21+1)}{2} = \frac{23 \times 22}{2} = 23 \times 11 = 253\).

Always check if the base expression can be simplified before calculating the number of terms, as simplification can change the type of expansion (e.g., from trinomial to binomial) and significantly affect the number of terms.