Problem 47 - Contemporary Precalculus

Question

Find the indicated term of the expansion of the given expression. fourth, \(\left(u^{-2}+\frac{u}{2}\right)^{7}\)

Step-by-Step Solution

Answer

Answer: \(\frac{35}{8}u^{-5}\)

Step 1: Apply the binomial theorem

Using the binomial theorem formula, we have: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k = \left(u^{-2}+\frac{u}{2}\right)^{7}=\sum_{k=0}^{7} \binom{7}{k}u^{-14+2k}\left(\frac{u}{2}\right)^k\).

Step 2: Find the fourth term

To find the fourth term in the expansion (corresponding to \(k=3\)), we will find the term of the expansion for \(k=3\). We have \(\binom{7}{3}u^{-14+2(3)}\left(\frac{u}{2}\right)^3\).

Step 3: Compute the binomial coefficient

To calculate \(\binom{7}{3}\), we use the formula \(\binom{7}{3} =\frac{7!}{3!(7-3)!} = \frac{7!}{3!4!}\). This gives \(\binom{7}{3} =\frac{7\times6\times5\times4!}{3\times2\times1\times4!} = \frac{7\times6\times5}{3\times2\times1}=7\times5=35\).

Step 4: Substitute the binomial coefficient into the term

Now we will substitute the value of \(\binom{7}{3}\) into our term: \(35u^{-14+2(3)}\left(\frac{u}{2}\right)^3=35u^{-8} \left(\frac{u}{2}\right)^3\).

Step 5: Simplify the term

Simplify the term further: \(35u^{-8} \times\frac{u^3}{8}= \frac{35}{8}u^{-8+3}=\frac{35}{8}u^{-5}\). The fourth term of the expansion of the given expression is \(\frac{35}{8}u^{-5}\).

Key Concepts

Binomial Coefficient

The binomial coefficient is a key element in combinatorics and algebra, representing the number of ways to choose a subset of items from a larger set, without concern for the order of selection. This concept is pivotal in the expansion of binomial expressions. For binomial coefficients, we use the notation \( \binom{n}{k} \), which is read as 'n choose k'.

The formula to compute the binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n! \) is the factorial of \( n \) and indicates the product of all positive integers up to \( n \) (e.g., \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)).

When simplifying \( \binom{n}{k} \), many factors in the numerator and denominator can cancel each other out, making it unnecessary to calculate the entire factorial. For example, \( \binom{7}{3} \) simplifies to \( 35 \), as seen in our exercise, without having to compute the entire value of \( 7! \).

Binomial Theorem Formula

The Binomial Theorem is a quick way to expand expressions that are raised to a power in the form of \( (a+b)^n \). It is a formula that allows us to write the expansion without having to multiply the binomial by itself repeatedly. The theorem states: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

The sum, denoted by \( \sum \) in the formula, runs from \( k=0 \) to \( k=n \) and tells us to add each term of the expansion together. Each term in the sum is a product of a binomial coefficient, \( a \) raised to a decreasing power, and \( b \) raised to an increasing power as \( k \) increases. In our exercise example, the expression \( (u^{-2} + \frac{u}{2})^7 \) is expanded using the binomial theorem to obtain the specific terms.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a process that combines like terms and reduces expressions to their simplest form. This involves several strategies such as combining like terms, using exponent rules, and doing arithmetic operations.

In our exercise, we simplified the term \( 35u^{-8} \times \frac{u^3}{8} \) after we found the fourth term in the binomial expansion. Simplification is done by adding exponents when multiplying powers of \( u \) together (as per the laws of exponents) and by performing division. The final result was \( \frac{35}{8}u^{-5} \) for the fourth term of the expression. This step is crucial, as it translates the formal expansion into a readily understandable and usable algebraic term.