1. Key Concept: The General Term of a Binomial Expansion

For a binomial expansion of the form $(a+b)^n$, the general term (or the $(r+1)^{th}$ term), denoted as $T_{r+1}$, is given by the formula: $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ where $n$ is the power of the binomial, and $r$ is an integer ranging from $0$ to $n$. This formula is crucial for finding specific terms in an expansion, such as the constant term, the term independent of $x$, or terms with a specific power of $x$.

2. Step-by-Step Solution

Let's apply this concept to the given problem: we need to find the constant term in the expansion of ${\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}$.

Step 1: Identify $a$, $b$, and $n$ Comparing the given expression with $(a+b)^n$:

• $a = \sqrt{x}$
• $b = -{k \over {{x^2}}}$ (Note the negative sign, which is part of $b$)
• $n = 10$

Step 2: Write down the General Term ($T_{r+1}$) Substitute $a$, $b$, and $n$ into the general term formula: $T_{r+1} = \binom{10}{r} \left(\sqrt{x}\right)^{10-r} \left(-{k \over {{x^2}}}\right)^r$

Step 3: Simplify the powers of $x$ Our goal is to combine all terms involving $x$ so we can determine its overall exponent.

• Recall that $\sqrt{x} = x^{1/2}$.
• Recall that $\frac{1}{x^2} = x^{-2}$.

Now, substitute these into the expression for $T_{r+1}$: $T_{r+1} = \binom{10}{r} \left(x^{1/2}\right)^{10-r} \left(-k \cdot x^{-2}\right)^r$ Apply the power rule $(a^m)^n = a^{mn}$ and $(ab)^n = a^n b^n$: $T_{r+1} = \binom{10}{r} x^{\frac{1}{2}(10-r)} (-k)^r (x^{-2})^r$ $T_{r+1} = \binom{10}{r} x^{\frac{10-r}{2}} (-k)^r x^{-2r}$

Step 4: Combine all terms involving $x$ Using the rule $a^m \cdot a^n = a^{m+n}$, combine the $x$ terms: $T_{r+1} = \binom{10}{r} (-k)^r x^{\frac{10-r}{2} - 2r}$ $T_{r+1} = \binom{10}{r} (-k)^r x^{\frac{10-r-4r}{2}}$ $T_{r+1} = \binom{10}{r} (-k)^r x^{\frac{10-5r}{2}}$

Step 5: Determine the condition for a constant term For a term to be "constant" (or independent of $x$), the variable $x$ must disappear from the term. This happens when the exponent of $x$ is equal to zero. So, we set the exponent of $x$ to zero: $\frac{10-5r}{2} = 0$

Step 6: Solve for $r$ Multiply both sides by 2: $10-5r = 0$ Add $5r$ to both sides: $10 = 5r$ Divide by 5: $r = 2$ This value of $r$ indicates that the constant term is the $T_{2+1} = T_3$ term (the 3rd term) in the expansion.

Step 7: Calculate the constant term Substitute $r=2$ back into the simplified general term expression from Step 4: $T_{2+1} = \binom{10}{2} (-k)^2 x^{\frac{10-5(2)}{2}}$ $T_3 = \binom{10}{2} (-k)^2 x^{\frac{10-10}{2}}$ $T_3 = \binom{10}{2} (-k)^2 x^0$ Since $x^0 = 1$ (for $x \neq 0$), the constant term is: $T_3 = \binom{10}{2} (-k)^2$ Now, calculate $\binom{10}{2}$: $\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10!}{2!8!} = \frac{10 \times 9}{2 \times 1} = 45$ Also, $(-k)^2 = k^2$. So, the constant term is $45k^2$.

Step 8: Equate the constant term to the given value and solve for $|k|$ The problem states that the constant term is 405. $45k^2 = 405$ Divide both sides by 45: $k^2 = \frac{405}{45}$ $k^2 = 9$ Taking the square root of both sides: $k = \pm\sqrt{9}$ $k = \pm 3$ The question asks for $|k|$. $|k| = |\pm 3|$ $|k| = 3$

3. Tips and Common Mistakes

• Sign Errors: Be very careful with the sign of the second term, $b$. In this case, $b = -k/x^2$. Forgetting the negative sign would lead to $(-k)^r$ becoming $k^r$, which could change the final sign if $r$ were odd. However, since $r=2$ (even), $(-k)^2 = k^2$, so the error might not be immediately apparent in this specific problem, but it's a critical point to remember.
• Exponent Simplification: Mistakes often occur when combining fractional and negative exponents. Always convert roots to fractional exponents (e.g., $\sqrt{x} = x^{1/2}$) and reciprocals to negative exponents (e.g., $1/x^2 = x^{-2}$) before combining.
• Understanding "Constant Term": The constant term is precisely the term where the variable (in this case, $x$) vanishes, meaning its exponent becomes zero.
• Absolute Value: Remember to address what the question asks for. If it asks for $k$, you'd provide $k=\pm 3$. If it asks for $|k|$, then the answer is the positive value.

4. Summary

To find the constant term in a binomial expansion, the key strategy is to first write down the general term $T_{r+1}$. Then, simplify all the variable terms ($x$ in this case) to get a single power of $x$. By setting this exponent to zero, you can solve for $r$, which identifies the position of the constant term. Finally, substitute this value of $r$ back into the non-$x$ part of the general term to calculate the constant term's value, and then solve for the unknown variable (k in this problem). This methodical approach ensures accuracy and clarity in solving such problems.