Question

Find the term which has the exponent of \(x\) as 8 in the expansion of \(\left(x^{\frac{1}{2}}-\frac{3}{x^{3} \sqrt{x}}\right)^{10}\). (1) \(\mathrm{T}_{2}\) (2) \(\mathrm{T}_{3}\) (3) \(\mathrm{T}_{4}\) (4) Does not exist

Short answer

Based on the step-by-step solution given, the problem solving required the use of the binomial theorem to find the term in the expansion of the provided binomial expression that had x raised to the power of 8. From the given equation, \(a = x^{\frac{1}{2}}\), \(b = -\frac{3}{x^{3} \sqrt{x}}\), and \(n = 10\) were identified as parameters for the binomial theorem formula. After deriving the general term and simplifying it, the value of \(r\) was found to be equal to 2 for a term with an \(x^8\) exponent. The term found to have x raised to the power of 8 was \(T_2=45\cdot \frac{9}{x^4}\), which is option (1) in the given choices.

Step-by-step solution

Write down the binomial theorem formula

To expand the given expression, we will use the binomial theorem formula, which is: \({(a+b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r}b^r}\), where \(a\) and \(b\) are the terms in the expression, \(n\) the exponent, and \(\binom{n}{r}\) is the binomial coefficient.

Identify the parameters in the given exercise

In this case, the given expression is \(\left(x^{\frac{1}{2}}-\frac{3}{x^{3} \sqrt{x}}\right)^{10}\). Comparing it with the binomial theorem formula, we can identify the values for a, b, and n as: \(a = x^{\frac{1}{2}}\) \(b = -\frac{3}{x^{3} \sqrt{x}}\) \(n = 10\)

Find the general term of the expansion

The general term of the expansion is given by \({\binom{n}{r} a^{n-r}b^r}\). We substitute the values for `a`, `b`, and `n` from Step 2. The general term becomes: \({T_r} = {\binom{10}{r} \left(x^{\frac{1}{2}}\right)^{10-r}\left(-\frac{3}{x^{3} \sqrt{x}}\right)^r}\)

Simplify the general term and find the exponent of x

Now, we will simplify the general term to identify the specific exponent of x: \({T_r} = {\binom{10}{r} x^{\frac{10}{2}-\frac{r}{2}}\left(-\frac{3}{x^{3} \sqrt{x}}\right)^r}\) In the rightmost term, \(\left(-\frac{3}{x^{3} \sqrt{x}}\right)^r\), we have \(x\) raised to the power of \(-(3r+r/2)\). So the overall exponent of x in \({T_r}\) will become: \(\frac{10}{2}-\frac{r}{2} -(3r+\frac{r}{2})\)

Find the term in which exponent of x is 8

We're looking for r such that the exponent of x is 8. Hence, set the overall exponent mentioned in Step 4 equal to 8 and solve for r: \(\frac{10}{2}-\frac{r}{2} -(3r+\frac{r}{2}) = 8\) Solve for r: \(r=2\)

Find the term with the exponent of x as 8

Now that we know r=2, plug r into the general term \({T_r}\) to find the term with x's exponent of 8: \(T_2 = \binom{10}{2} \left(x^{\frac{1}{2}}\right)^{8}\left(-\frac{3}{x^{3} \sqrt{x}}\right)^{2}\) \(T_2=45\cdot x^{4}\cdot\left(-\frac{3}{x^{3} \sqrt{x}}\right)^{2}\) \(T_2=45\cdot x^{4}\cdot\frac{9}{x^8}\) \(T_2=45\cdot \frac{9}{x^4}\) And the correct term with the exponent of x as 8 is \(T_2\), which is the (1) choice from the given options.

Key concepts

Binomial Theorem Formula

Understanding the Binomial Theorem Formula is vital for expanding expressions of the form \(a+b)^n\). It's essentially a way to express the expansion of a binomial raised to a power without having to multiply it out the long way. Here's the general formula: \[ (a+b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r}b^r \]
To break this down, the symbol \(\sum\) represents the sum over all terms from \(r=0\) to \(n\). The binomial coefficient \(\binom{n}{r}\) is a crucial part of this formula, often read as 'n choose r', and it determines how many ways you can choose \(r\) elements from a set of \(n\) elements. The terms \(a^{n-r}\) and \(b^r\) indicate the powers that \(a\) and \(b\) are raised to, and they change with each term of the expansion. When you apply this formula, it's a systematic approach to obtaining the full expanded form of a binomial expression.

General Term of Binomial Expansion

In the context of binomial expansion, the general term refers to the \(r\)th term in the expansion, which can be written as \(T_{r+1} = \binom{n}{r} a^{n-r}b^r\). The 'general' part means that this term can represent any term in the expansion, depending on the value of \(r\).

The term operates like a template. By plugging in different values of \(r\), ranging from 0 to \(n\), you produce each of the terms in the binomial expansion. It's particularly helpful when looking for a specific term, such as the one with the exponent of \(x\) as 8 in our exercise. By identifying that specific value for \(r\), you can use the general term to pull out just the piece of the expansion that interests you, without having to expand the entire thing.

Solving Binomial Coefficients

Solving binomial coefficients often intimidates students, but it's a straightforward process. The binomial coefficient \(\binom{n}{r}\), which appears in both the binomial theorem formula and the general term of the expansion, stands for the number of combinations of \(n\) items taken \(r\) at a time and can be calculated using the formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \(n!\) denotes 'n factorial', which means the product of all positive integers up to \(n\). To find a particular binomial coefficient, simply substitute your values for \(n\) and \(r\) and carry out the arithmetic. For example, the coefficient \(\binom{10}{2}\) means 'how many ways can you choose 2 out of 10 items?' By computing it, you obtain 45, indicating there are 45 different pairs you could form from 10 distinct items.

Exponent Calculation in Polynomials

Exponent calculation is indispensable when dealing with polynomials, especially in the context of binomial expansions. Understanding how to manage exponents when they are added, subtracted, or multiplied is key to simplifying expressions. In polynomials arising from binomial expansions, each term involves exponents that result from the formula applied to \(a\) and \(b\), as seen with phrases like \(a^{n-r}\) and \(b^r\).

For example, to find the term with an exponent of \(x\) as 8, we align the sum of the exponents from each part of the term \(T_r\) to equal 8, which is fundamentally solving an equation. We determine the value of \(r\) that will make the exponent of \(x\) in \(T_r\) turn out to be 8. This task involves understanding the laws of exponents, such as knowing that multiplying powers adds exponents and dividing powers subtracts them. Knowing these rules allows us to simplify the general term to the point where we can clearly see what value of \(r\) gives us the desired power of \(x\), as demonstrated in our exercise.