How do you find the eight term in the expansion (a + b)^14?

Apr 10, 2016

$3432 {a}^{7} {b}^{7}$

Explanation:

If you have not had much to do with Binomial Expansion, Khan Academy have a great video that simplifies the process:

Consider the formula used to expand a Binomial Equation:
${\sum}_{k = 0}^{n} {\cdot}^{n} {C}_{k} \cdot {a}^{n - k} \cdot {b}^{k}$

Where:
$n$ is the power to which the equation is raised.

Therefore, if we consider your equation:
${\left(a + b\right)}^{14}$

From your equation, we can see that the power to which the equation is raised to is: 14.

Before we have a look at the 8th term, let's have a look at the first term so we can observe the relationship between the Binomial Theory and the subsequent expansion:

For the first term of: ${\left(a + b\right)}^{14}$

${\sum}_{k = 0}^{n} {\cdot}^{n} {C}_{k} \cdot {a}^{n - k} \cdot {b}^{k}$
${=}^{14} {C}_{0} \cdot {a}^{14} \cdot {b}^{0}$
$= 1 \cdot {a}^{14} \cdot 1$
$= {a}^{14}$

Now let's have a look at expanding the 8th Term of the equation:

${\sum}_{k = 0}^{n} {\cdot}^{n} {C}_{k} \cdot {a}^{n - k} \cdot {b}^{k}$
${=}^{14} {C}_{7} \cdot {a}^{7} \cdot {b}^{7}$
$= 3432 \cdot {a}^{7} \cdot {b}^{7}$
$= 3432 {a}^{7} {b}^{7}$