Finding the Constant Term in Binomial Expansion

Let's analyze the given expression for binomial expansion: \({\left( {\sqrt x - \frac{k}{{{x^2}}}} \right)^{10}}\). This is in the form \((a+b)^n\), where \(a = \sqrt x = x^{1/2}\), \(b = -\frac{k}{x^2} = -k x^{-2}\), and the power of the expansion is \(n = 10\).

The general term in the expansion of \((a+b)^n\) is given by the formula:

\(T_{r+1} = {^nC_r a^{n-r} b^r}\)

Let's substitute the values from our expression into this formula:

\(T_{r+1} = {^{10}C_r (\sqrt x)^{10-r} \left(-\frac{k}{x^2}\right)^r}\)

Now, let's simplify the terms involving 'x':

\(T_{r+1} = {^{10}C_r (x^{1/2})^{10-r} (-1)^r k^r (x^{-2})^r}\)

\(T_{r+1} = {^{10}C_r x^{(1/2)(10-r)} (-1)^r k^r x^{-2r}}\)

\(T_{r+1} = {^{10}C_r x^{5 - r/2} (-1)^r k^r x^{-2r}}\)

Combine the powers of 'x' using the rule \(x^m \cdot x^n = x^{m+n}\):

\(T_{r+1} = {^{10}C_r (-1)^r k^r x^{5 - r/2 - 2r}}\)

\(T_{r+1} = {^{10}C_r (-1)^r k^r x^{5 - 5r/2}}\)

Finding the Value of r for the Constant Term

A constant term in a binomial expansion is a term that does not contain the variable 'x'. This means the power of 'x' in that term must be zero.

Setting the power of 'x' from our general term equal to zero:

\(5 - \frac{5r}{2} = 0\)

Solving for 'r':

• Add \(\frac{5r}{2}\) to both sides: \(5 = \frac{5r}{2}\)
• Multiply both sides by 2: \(10 = 5r\)
• Divide both sides by 5: \(r = \frac{10}{5}\)
• So, \(r = 2\)

This tells us that the term independent of x (the constant term) is the term where \(r=2\), which is the \((2+1)\)th or 3rd term (\(T_3\)) in the expansion.

Calculating the Constant Term

Substitute the value of \(r=2\) back into the simplified general term formula:

Constant Term \( = {^{10}C_2 (-1)^2 k^2 x^{5 - 5(2)/2}}\)

Constant Term \( = {^{10}C_2 (1) k^2 x^{5 - 5}}\)

Constant Term \( = {^{10}C_2 k^2 x^0}\)

Constant Term \( = {^{10}C_2 k^2}\)

Evaluating the Binomial Coefficient

Now, we calculate the binomial coefficient \(^{10}C_2\):

\(^{10}C_2 = \frac{10!}{2!(10-2)!} = \frac{10!}{2!8!}\)

\(^{10}C_2 = \frac{10 \times 9 \times 8!}{2 \times 1 \times 8!} = \frac{10 \times 9}{2}\)

\(^{10}C_2 = \frac{90}{2} = 45\)

So, the constant term is \(45k^2\).

Solving for k

We are given that the constant term in the expansion is 405.

Therefore, we can set up the equation:

\(45k^2 = 405\)

To find \(k^2\), divide both sides by 45:

\(k^2 = \frac{405}{45}\)

Performing the division:

\(k^2 = 9\)

To find the value(s) of k, take the square root of both sides:

\(k = \pm \sqrt{9}\)

\(k = \pm 3\)

Thus, the possible values of k are \(+3\) and \(-3\).

Revision Table: Binomial Expansion Concepts

Additional Information: Properties of Exponents

When dealing with terms like \(\sqrt x\) and \(\frac{1}{x^2}\) in binomial expansion problems, it's important to remember exponent rules:

• Square root: \(\sqrt x = x^{1/2}\)
• Reciprocal: \(\frac{1}{x^n} = x^{-n}\)
• Power of a power: \((x^m)^n = x^{mn}\)
• Multiplying powers with the same base: \(x^m \cdot x^n = x^{m+n}\)

Using these rules helps in simplifying the power of the variable 'x' in the general term, which is crucial for finding the constant term or any specific term based on the power of 'x'. The coefficient \(^nC_r\) and the constant factor from the terms 'a' and 'b' determine the value of the term once 'r' is found.