Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the specified \(n\) th term in the expansion of the binomial. \((5 a+6 b)^{5}, n=5\)

Step-by-Step Solution

Answer

Thus, the fifth term in the binomial expansion of \((5a + 6b)^5\) is \(32,400ab^4\).

Step 1: Identify the binomial and its exponent

The binomial in this problem is \(5a + 6b\) and it's raised to the power of 5: \((5a + 6b)^5\). We are looking for the 5th term in the expansion of this binomial.

Step 2: Apply the binomial theorem

When using the binomial theorem to find a particular term in a binomial expansion, a formula can be applied: \(T_{r+1} = ^nC_r * (a^{n-r}) * (b^r)\). Here, r represents the term number minus 1, n is the power of the binomial, a and b are the first and second terms of the binomial respectively. For this problem, n=5, r=4 (because we're finding the 5th term so r+1=n), a=5a, and b=6b.\nCalculating the values, we get: \(T_{5} = ^5C_4 * (5a^{5-4}) * (6b^4)\)

Step 3: Calculate the result

Use the formula for binomial coefficient, \(^nC_r\) = n! / [r!(n-r)!]. For this problem, \(^5C_4 = 5!/4!(5-4)! = 5\). Now we calculate the value of the term as follows: \(T_{5} = 5 * (5a^{5-4}) * (6b^4) = 5* 5a * 1296b^4 = 5*5* 1296 * a*b^4 = 32,400ab^4\)

Step 4: Verify the result

The 5th term in the expansion of the binomial \((5a + 6b)^5\) that we calculated using the binomial theorem and coefficient formula is 32,400ab^4

Key Concepts

Binomial Expansion

A binomial expansion refers to expressing a binomial, like \((5a + 6b)^5\), into a sum of terms. Each term consists of contributions from each part of the binomial raised to increasing powers. For instance, the expansion of \((a + b)^2\) is \(a^2 + 2ab + b^2\). In general, the binomial theorem provides a formula for expanding any binomial raised to a power.

• Helps in breaking down complex expressions.
• Useful in simplifying calculations in algebra.

In this case, we sought the 5th term of \((5a+6b)^5\). Using the binomial expansion, you pick terms where powers of \(5a\) decrease by one while powers of \(6b\) increase by one.

Binomial Coefficients

Binomial coefficients are essential to determining each term's value in a binomial expansion. These coefficients, \(^nC_r\), represent the number of ways you can choose \(r\) successes from \(n\) trials, and are found in Pascal's triangle.

• Calculated using factorials: \(^nC_r = \frac{n!}{r!(n-r)!}\).
• Increase predictably in each expansion row.

In our problem, we found \(^5C_4\) to determine the coefficient of the 5th term in the binomial expansion, which is 5.

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It's a gateway to exploring more complex math concepts and solving problems ranging from simple equations to intricate polynomial expressions. Making use of algebra, we:

• Set up equations using symbols like \(a\) and \(b\).
• Applied arithmetic operations to simplify expressions.
• Used algebraic identities, like the binomial theorem, to expand and simplify polynomials.

In the solution process for our problem, algebraic concepts were applied to calculate specific terms and coefficients, aiding in understanding how complex expressions break down into simpler components.