Integrated Math 2–3 Honors — Rationalizing Radical Denominators with Fractional Exponents
Corrected lesson transcript


## 0:00 — Rationalizing Radical Denominators


## 0:05 — Review: Square-Root Denominators

[0:05] We talked Tuesday about dividing radicals.
[0:12] We didn't really divide.
[0:13] What did we do?
[0:15] Multiply.
[0:16] We multiplied.
[0:17] and the whole goal was somebody said it,
[0:21] Rationalize the denominator.
[0:23] So square root of two is an irrational number.
[0:27] And so what are we gonna multiply by?
[0:31] Square root two over square root two.
[0:35] Yes, square root three times square root two over two.
[0:40] And we're done.
[0:41] We rationalize the denominator.
[0:43] Simplify, nothing else simplifies.
[0:45] Easy peasy, right?

## 0:46 — Why One Cube-Root Factor Is Not Enough

[0:46] Is the cube root of two irrational?
[0:51] Yes. Unless the radicand is a perfect cube, it is irrational.
[1:02] If I multiply by cube root two over cube root two, I get cube root two over cube root four.
[1:09] root of 2 over the cube root of 4.
[1:16] Is the cube root of 4 rational?
[1:20] No.
[1:23] So now what?
[1:28] Okay, I like that.
[1:37] Now we get cube root four over cube root eight.
[1:44] And what is the cube root of eight?
[1:48] Now we have a rational denominator.
[1:51] That would be the answer.

## 1:55 — Choose the Missing Power

[1:55] We could have done it all in one step, instead of doing it twice.
[2:09] What should we have multiplied by?
[2:12] The cube root of four.
[2:18] Also known as the cube root of two squared.
[2:27] That gives us the cube root of two cubed.
[2:30] The result is cube root four over two.

## 2:37 — A Cube-Root Denominator with a Power

[2:37] Let's try another one.
[2:44] No.
[2:45] Oh, let's try one.
[3:00] The denominator also contains five squared.
[3:03] So what am I gonna multiply here?
[3:16] Just the cube root of five.
[3:20] The denominator becomes the cube root of five cubed.
[3:26] Oops, still got a cube root here.
[3:31] The cube root of five cubed is just five.
[3:33] Let's try another one.

## 3:40 — Generalize to Any Root Index

[3:40] So
[3:42] For the seventh root of two cubed, what do we need?
[3:50] We need two to the fourth power.
[4:05] Two cubed times two to the fourth is two to the seventh.
[4:12] The seventh root of two to the seventh is two.
[4:26] We add these exponents to reach the root index.
[4:33] So we choose the fourth power of that base.

## 4:36 — Use Fractional Exponents

[4:36] We could write this as the tenth root of four cubed.
[4:56] What power of four do we need this time?
[5:00] Seven tenths.
[5:10] And that's for the tenth root four to the tenth
[5:20] We do not need radical notation; we can keep fractional exponents.
[5:30] What power adds to three tenths to get one?
[5:36] Seven tenths.
[5:40] Multiply by four to the seven tenths over itself.
[5:43] Four to the three tenths times four to the seven tenths is four to the first power.
[5:52] If you wanted to write it in radical form, you could.
[5:55] In radical form, the numerator is the tenth root of four to the seventh power.
[5:58] over four.

## 5:58 — Fill to the Next Whole Exponent

[5:58] We could try to reach the first power, but that would require negative five thirds.
[6:04] five thirds?
[6:11] That would leave a negative exponent in the numerator, which we do not want.
[6:16] So we fill the denominator exponent to the next whole number.
[6:20] Eight divided by three is what?
[6:22] Two and two thirds.
[6:24] The next whole exponent is three.
[6:29] So we need two to the what power?
[6:37] One third.
[6:38] Eight thirds plus one third is nine thirds, or three.
[6:59] The numerator can be written as the cube root of two.
[7:06] Like that.
[7:08] Or you could leave it like that.
[7:09] Did I write that incorrectly?
[7:12] Sorry, that exponent is three.
[7:13] Thank you.

## 7:13 — A Larger Fractional Exponent

[7:13] Thirty sevenths is four and something.
[7:18] The next whole exponent is five, or thirty-five sevenths.
[7:29] So we multiply by three to a power.
[7:35] To get from thirty to thirty-five, we need five sevenths.
[7:47] The result is three to the five sevenths over three to the fifth.
[7:55] If you did want to write it in radical form, it would be what?
[7:58] In radical form, the numerator is the seventh root of three to the fifth.
[8:03] Whatever number that comes out to be.
[8:05] Or we can leave it like that.

## 8:05 — Rationalize with Variables

[8:05] Let's make it x.
[8:07] Okay.
[8:10] Did it matter what this number was?
[8:14] No.
[8:16] We use the exact same method with variables.
[8:19] What power of x do we need?
[8:22] Three fifths. Two fifths plus three fifths is five fifths.

## 8:34 — Remove Negative Fractional Exponents

[8:34] So on the warm up we did the other day, I didn't really explain that very well or at
[8:39] all.
[8:41] First, we do not want negative exponents.
[8:46] For x to the negative two fifths, first rewrite it as one over x to the positive two fifths.
[8:57] Then rationalize so there are no fractional exponents in the denominator.
[9:02] it's kind of the same thing as rationalizing the denominator let's try

## 9:15 — Final Variable Example

[9:15] We want no negative exponents and no fractional exponents in the denominator.
[9:21] First rewrite the negative exponent as a reciprocal.
[9:30] That gives one over x to the positive eleven fourths; then remove the fractional exponent from the denominator.
[9:50] So how much do we need?
[9:52] One fourth.
[9:59] Eleven fourths plus one fourth is twelve fourths, or three.
[10:05] We will just like that.
[10:09] There are no negative exponents
[10:11] and no fractional exponents in the denominator.
[10:17] Cool?
[10:18] Okay.
[10:19] That was the last lesson before our test early next week.