Understanding Binomial Expansion and the Independent Term

The question asks for the term that does not contain the variable 'x' in the binomial expansion of \( \left( \frac {2}{x^2} - \sqrt x \right)^{10} \). This term is often called the "term independent of x". To find this term, we need to use the general formula for the terms in a binomial expansion and identify the term where the power of 'x' is zero.

General Term Formula for Binomial Expansion

The binomial theorem states that the expansion of \( (a+b)^n \) is given by:

\[ (a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]

The general term, or the \((r+1)^{\text{th}}\) term, in the expansion is given by:

\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]

In our given expression \( \left( \frac {2}{x^2} - \sqrt x \right)^{10} \):

• \(a = \frac{2}{x^2} = 2x^{-2}\)
• \(b = -\sqrt x = -x^{1/2}\)
• \(n = 10\)

Finding the Power of x in the General Term

Let's substitute the values of a, b, and n into the general term formula:

\[ T_{r+1} = \binom{10}{r} \left( 2x^{-2} \right)^{10-r} \left( -x^{1/2} \right)^r \]

Now, let's simplify the expression to find the total power of x in the term:

\[ T_{r+1} = \binom{10}{r} (2)^{10-r} (x^{-2})^{10-r} (-1)^r (x^{1/2})^r \]

\[ T_{r+1} = \binom{10}{r} 2^{10-r} x^{-2(10-r)} (-1)^r x^{r/2} \]

\[ T_{r+1} = \binom{10}{r} 2^{10-r} (-1)^r x^{-20 + 2r} x^{r/2} \]

Combining the terms with x, we add their exponents:

\[ T_{r+1} = \binom{10}{r} 2^{10-r} (-1)^r x^{(-20 + 2r + r/2)} \]

To simplify the exponent of x:

\[ -20 + 2r + \frac{r}{2} = -20 + \frac{4r}{2} + \frac{r}{2} = -20 + \frac{5r}{2} \]

So, the general term is:

\[ T_{r+1} = \binom{10}{r} 2^{10-r} (-1)^r x^{-20 + \frac{5r}{2}} \]

Determining the Value of r for the Independent Term

For the term to be independent of x, the exponent of x must be equal to zero. Therefore, we set the exponent to 0 and solve for r:

\[ -20 + \frac{5r}{2} = 0 \]

\[ \frac{5r}{2} = 20 \]

\[ 5r = 20 \times 2 \]

\[ 5r = 40 \]

\[ r = \frac{40}{5} \]

\[ r = 8 \]

Since r must be an integer and \(0 \le r \le n\), \(r=8\) is a valid value.

Calculating the Value of the Independent Term

The term independent of x is the \((r+1)^{\text{th}}\) term with \(r=8\), which is the \((8+1)^{\text{th}}\) or 9th term. We substitute \(r=8\) into the general term formula (excluding the x term):

The term is \( \binom{10}{8} 2^{10-8} (-1)^8 \)

Calculate the components:

• \( \binom{10}{8} \): This is the binomial coefficient. \( \binom{10}{8} = \binom{10}{10-8} = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = \frac{90}{2} = 45 \)
• \( 2^{10-8} = 2^2 = 4 \)
• \( (-1)^8 = 1 \) (since the exponent is an even number)

Now, multiply these values together to get the value of the term independent of x:

Value = \( 45 \times 4 \times 1 = 180 \)

The term independent of x in the binomial expansion of \( \left( \frac {2}{x^2} - \sqrt x \right)^{10} \) is 180.

Revision Table: Key Concepts

Additional Information: Finding Specific Terms in Binomial Expansion

Finding the term independent of x is a common application of the binomial theorem. Here are related concepts:

• Middle Term(s): In the expansion of \((a+b)^n\), if n is even, there is one middle term (the \((n/2 + 1)^{\text{th}}\) term). If n is odd, there are two middle terms (the \((n+1)/2^{\text{th}}\) and \((n+3)/2^{\text{th}}\) terms).
• Term with a Specific Power of x: Similar to finding the term independent of x, you can find the term with a specific power of x (\(x^k\)) by setting the exponent of x in the general term to k and solving for r.
• Sum of Coefficients: The sum of coefficients in the expansion of \((a+b)^n\) is found by substituting a=1 and b=1 into the expression, which gives \((1+1)^n = 2^n\).
• Relationship between Terms: The ratio of consecutive terms \( T_{r+1}/T_r \) can be used to find the greatest term or the term with the greatest coefficient.

Understanding the general term formula is crucial for solving various problems related to binomial expansions.