Using the Expansion: Surds, Estimates and Identities

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle first. Take root three plus root two, and raise it to the power six. Then take root three minus root two, also to the power six. Both are irrational numbers, right? Now add them. What do you think you get? Some messy number full of roots? No! You get a plain whole number. How can the roots simply vanish? Today we will see why, and we will also compare two huge numbers without a calculator. Questions like these come up often in JEE, so let's begin.

0:36Conjugates: half the terms cancel

So, look at the screen. The tick-shaped sign is the square root sign, so this is root b. One number is a plus root b, the other is a minus root b. Same numbers, only the sign in the middle is different. We call them a conjugate pair. Okay? Now expand a plus root b, to the power n, where n is a positive integer. We get a to the power n, plus n C one, a to the power n minus one, root b, plus n C two, a to the power n minus two, root b squared, plus n C three, a to the power n minus three, root b cubed, and the three dots mean, and so on. The C with a small n at its top left is the binomial coefficient. With me? Now, a minus root b, to the power n. Root b becomes minus root b. So what happens to each term? Yes, the signs alternate! Minus root b to an odd power stays negative, and to an even power it turns positive. So the odd powers of root b pick up a minus sign, and the even powers keep their plus. Right? Now the magic. Add these two lines. Which terms survive? Yes, exactly! The odd powers of root b cancel: one plus, one minus. The even powers appear twice, so they double. And root b squared is just b, so the roots disappear! Clear? So, the first useful fact. The sum, a plus root b to the power n, plus a minus root b to the power n, has only even powers of root b. So it is an integer, when a and b are integers. Remember that condition, okay? And if we subtract instead? Now the even powers cancel, and the odd powers double. Each odd power of root b is root b times a power of b. So, again for integers a and b, the difference is an integer multiple of root b. Got it?

2:36Estimates: the first few terms are enough

Now the second use: estimates. When x is small, x squared and x cubed are smaller still. So the first two or three terms are enough. One plus x, to the power n, is approximately one, plus n x, plus n times n minus one, over two, x squared. The wavy equals sign means approximately equal. Okay? And here is an exact inequality. If x is greater than zero, and n is greater than or equal to two, then one plus x, to the power n, is greater than one plus n x. Why? Every term after the first two is positive, and we dropped them. Right? So, here is one from the course. Which is larger: one point zero one, to the power one million, or ten thousand? One point zero one is hardly more than one, isn't it? Oh no, careful! If you said ten thousand, that is the trap. One point zero one is one plus x, with x equal to zero point zero one, which is positive. So it is greater than one, plus ten to the power six, that is one million, times zero point zero one. That is ten thousand and one, more than ten thousand. So one point zero one, to the power one million, is larger. No calculator needed! Clear?

3:58Example 1: find (root 2 + 1)^6 + (root 2 - 1)^6

Alright, example one. Find root two plus one, to the power six, plus root two minus one, to the power six. Pause the video and try it. Done? Okay, let's check. Root two plus one and root two minus one are a conjugate pair. So the terms with odd powers of the second number, one, cancel, and the terms with even powers double. Two, times, in square brackets: root two to the power six, plus six C two, root two to the power four, plus six C four, root two squared, plus one. With me? Now, six C two and six C four are both fifteen. Root two to the power six is eight, to the power four is four, and squared is two. So: eight, plus fifteen times four, that is sixty, plus fifteen times two, that is thirty, plus one. Okay? Add them: eight plus sixty plus thirty plus one, ninety nine. So the answer is ninety nine, right? No, no, wait! Look at the two outside the square brackets. Every surviving term comes twice, once from each bracket. So, two times ninety nine, and the cross means times: one hundred and ninety eight. Just a whole number, from two irrational ones. Perfect!

5:20Example 2: find (root 3 + root 2)^6 + (root 3 - root 2)^6

Now, our puzzle from the start. Find root three plus root two, to the power six, plus root three minus root two, to the power six. Pause the video and try it. Done? Okay, let's check. Again a conjugate pair, so the odd powers of root two cancel, and the even powers double. Two, times, in square brackets: root three to the power six, plus six C two, root three to the power four, root two squared, plus six C four, root three squared, root two to the power four, plus root two to the power six. Clear? But wait. Our fact needed a to be an integer, and root three is not one. So, will a root survive here? No! Look at the powers of root three: six, then four, then two. All even! So root three to the power six is twenty seven, to the power four is nine, and squared is three. Root two squared is two, to the power four is four, and to the power six is eight. The dots here mean times. Okay? Fifteen times nine times two is two hundred and seventy, and fifteen times three times four is one hundred and eighty. With twenty seven and eight, the bracket is four hundred and eighty five. Times two: nine hundred and seventy. There it is, the whole number from our puzzle! Very good! Now a topper's trick, to check it a second way. The power six is a square, then a cube. So, square first. Root three plus root two, squared, is three plus two plus two root six: five plus two root six. The plus-or-minus sign means the same works with minus: root three minus root two, squared, is five minus two root six. Right? Now cube. Five plus two root six, cubed, is four hundred and eighty five, plus one hundred and ninety eight root six. And the conjugate, five minus two root six, cubed, gives the same numbers, but minus one hundred and ninety eight root six. Got it? Add the two, and the root six parts cancel. Four hundred and eighty five, plus four hundred and eighty five: nine hundred and seventy. Same answer, and much faster. That's it!

7:45Quick revision: conjugate sums, differences and estimates

So, quick revision. Conjugate sum: the odd powers of root b cancel, and the even powers double, so the roots vanish. Conjugate difference: an integer multiple of root b, again when a and b are integers. And for estimates, keep the first few terms: one plus x, to the power n, is greater than one plus n x, when x is positive and n is at least two. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!