### Video Transcript

Using the binomial theorem, approximate to three decimal places the value of 1.05 to the sixth power.

We begin by recalling that one version of the binomial theorem states that one plus π₯ to the πth power is equal to one plus ππ₯ plus π multiplied by π minus one multiplied by π₯ squared over two factorial plus π multiplied by π minus one multiplied by π minus two multiplied by π₯ cubed over three factorial and so on. We note that this version of the theorem only holds when the absolute value of π₯ is less than one. This means that π₯ must be greater than negative one and less than one. As we are trying to raise one plus π₯ to the πth power, it allows us to calculate this providing π₯ plus one is greater than zero and less than two.

In this question, π₯ plus one is equal to 1.05. If one plus π₯ is equal to 1.05, then subtracting one from both sides, we have π₯ is equal to 0.05. We are therefore trying to calculate one plus 0.05 raised to the sixth power. Our expansion becomes one plus six multiplied by 0.05 plus six multiplied by five multiplied by 0.05 squared divided by two factorial plus six multiplied by five multiplied by four multiplied by 0.05 cubed divided by three factorial plus six multiplied by five multiplied by four multiplied by three multiplied by 0.05 to the fourth power divided by four factorial and so on. Six multiplied by 0.05 is 0.3. Since two factorial is equal to two, we can divide the numerator and denominator of our third term by two, leaving us with three multiplied by five multiplied by 0.05 squared. Typing this into the calculator gives us 0.0375.

We know that three factorial is equal to three multiplied by two multiplied by one which equals six. This means that the next term simplifies to one multiplied by five multiplied by four multiplied by 0.05 cubed. This is equal to 0.0025. Four factorial is equal to 24. So, the next term simplifies to five multiplied by three multiplied by 0.05 to the fourth power. This is equal to 0.00009375. At this stage, it is worth recalling that we want to give our answer to three decimal places. The value of this term to three decimal places will be zero, and the same is true for the two terms that follow this. As a result, we can ignore these terms when approximating our value to three decimal places. Our expression is therefore equal to one plus 0.3 plus 0.0375 plus 0.0025. This is equal to 1.34.

Using the binomial theorem, the value of 1.05 to the sixth power approximated to three decimal places is 1.34 or 1.340. We could check this answer by simply typing 1.05 to the sixth power into our calculator. This is equal to 1.3400956 and so on, which rounds to 1.34 to three decimal places.