Find the third term in the
expansion of two 𝑥 plus five over the square root of 𝑥 to the power of five.
This is an example of a binomial
expansion written in the form 𝑎 plus 𝑏 to the power of 𝑛. We could write out the whole
expansion. However, we know that the general
term 𝑎 sub 𝑟 plus one is equal to 𝑛 choose 𝑟 multiplied by 𝑎 to the power of 𝑛
minus 𝑟 multiplied by 𝑏 to the power of 𝑟. In this question, we want to find
the third term 𝑎 sub three. Inside the parentheses, we have two
terms. 𝑎, the first term, is equal to two
𝑥. And 𝑏, the second term, is equal
to five over root 𝑥.
The exponent or power that this is
raised to is five. Therefore, 𝑛 is equal to five. As we are trying to find the third
term, 𝑟 plus one is equal to three. Subtracting one from both sides of
this equation gives us 𝑟 is equal to two. We can now substitute these four
values into the general term formula. The third term is therefore equal
to five choose two multiplied by two 𝑥 cubed multiplied by five over the square
root of 𝑥 squared.
We know that 𝑛 choose 𝑟 is equal
to 𝑛 factorial divided by 𝑛 minus 𝑟 factorial multiplied by 𝑟 factorial. Five choose two is therefore equal
to five factorial divided by three factorial multiplied by two factorial. We can rewrite five factorial as
five multiplied by four multiplied by three factorial. This simplifies to five multiplied
by four divided by two factorial, which equals 10. Five choose two is equal to 10.
When cubing two 𝑥, we can cube the
two and 𝑥 separately. As two cubed is equal to eight, two
𝑥 cubed is equal to eight 𝑥 cubed. We can use a similar method when
squaring a fraction. We square the numerator and
denominator separately. Five squared is equal to 25, and
the square root of 𝑥 squared is 𝑥. A square rooting and squaring are
inverse operations. The third term is therefore equal
to 10 multiplied by eight 𝑥 cubed multiplied by 25 over 𝑥.
This can be simplified to 2000𝑥
squared as 25 multiplied by eight multiplied by 10 is 2000 and 𝑥 cubed divided by
𝑥 is 𝑥 squared. The third term in the expansion of
two 𝑥 plus five over the square root of 𝑥 to the power of five is 2000𝑥
squared. To save time, we could’ve also
calculated five choose two on the calculator.