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Simplify Rational Exponents

  • To simplify radical expressions we often split up the root over factors. If we are working with a square root, then we split it up over perfect squares.
  • Since radicals follow the same rules as exponents, we can use the quotient rule to split up radicals over division.
  • The square root of a squared number is always nonnegative.

Simplify Rational Exponents

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Objectives 0:07
  • Simplify Rational Exponents 0:25
    • Product Rule for Radicals
    • Product Rule to Simplify Square Roots
    • Quotient Rule for Radicals
    • Applications of Product and Quotient Rules
    • Higher Roots
  • Example 1 3:39
  • Example 2 6:35
  • Example 3 8:41
  • Example 4 11:09

Transcription: Simplify Rational Exponents

Welcome back to

In this lesson we are going to look at simplifying rational exponents.0003

The things we will look at are our product rule, quotient rule and what to do when they involve things like radicals.0009

We will also look at a few radicals that involve variables and how can we get into simplifying much higher roots.0018

If you have two non negative real numbers call them A and B.0028

And what you can do with them when they are being multiplied underneath a radical is put them under the same radical.0033

If those are already underneath the same radical and you are also free to go and separate them back out.0041

They are each underneath their own radical.0047

This rule right here is good for either simplifying or combining things.0050

You can use it in both direction, combine or break apart.0055

One very important thing to note is that this rule only works for multiplication,0060

do not try and use any type of splitting thing if you have addition or subtraction.0065

Where the simplifying part comes into play is by taking a very large number, large expression and breaking it up into those individual parts.0075

We can consider a radical simplified when there is no perfect square under the radical anymore.0085

Look at these different radicals, find the perfect squares that are in there and split up the roots over those perfect squares.0093

In a similar rule, we have that we can either combine things under radical when we are dealing with division0106

or if they are both under division, we can split up over the top and over the bottom.0112

This works as long as the value on the bottom is not equal to 0.0118

Of course that is because we cannot divide by 0.0122

Feel free to use this rule in both directions either combine or separate them.0127

Note that this rule only works for division.0131

Again, do not try and use this for addition or subtraction.0133

If we happen to have a variable underneath the root and it is raised to the power,0140

we can assume that variable represents a non negative number.0143

That will make sure that we are dealing with real numbers and not imaginary numbers.0149

The product in the quotient rule that we have just covered apply when all the variables up here under a radical sign as well.0155

We are dealing with non negative real numbers with these various different rules.0162

You can even use these rules for some much higher roots and when you are looking to simplify those it depends on the index.0171

For example, if you are looking to simplify a cubed root, then what we are doing is0183

we are trying to take out all the cubes that are underneath that root.0187

Here is a quick example.0190

Maybe we are looking at the 3√a3 without simplifying it to just an (a).0192

In cube roots this happens to be true whether (a) is positive or negative.0198

As long as you are dealing with square roots, the square root of the square number is always non negative.0204

The product and quotient will be applied to both our higher roots as well as just square roots.0212

Let us practice a few of these rules with some example questions.0221

If you want to simplify some numbers like the √4/49 feel free to split it up over the top and or the bottom.0225

And then you can take each of these separately.0234

This would be 2/7.0237

Here us one where they already have a root over the top and the bottom but I cannot simplify them individually.0245

I’m going to use that the quotient rule in the other direction and we will put them underneath 1 root.0251

From there I can figure out how many times 3 goes into 48.0259

I have that 3 goes in the 48 at least 10 times.0269

I have 18 left over, that would be 3 × 6.0277

By combining them underneath one root it allowed to go ahead and simplify the numbers a little bit better.0291

I just have to look at √16 is just 4.0296

Whether you are putting things underneath the same root or separating them out sometimes they cannot be simplified.0302

In this next one I will look at the root of the top and the root of the bottom.0311

With the bottom I can take the √36 is just 6 but the √ 5 has to stay as it is because it is not perfect square.0316

Let us look at one more.0331

You want to look at the √3/8 × √7/20335

With this one I have two different radicals.0347

Do not let those fractions distract you we are looking to combine them underneath the same radical.0351

I will just combine them together using multiplication.0359

Once I do that I could go ahead and continue simplifying from here.0365

This will be √21/60 and on the bottom 16 is a square number so I can even go further.0368

√21 / √16 will be √21 / 40379

All of these examples are designed to get you more familiar with either taking that root0386

and putting over both the parts or combining them into a single root.0390

Let us try a few that have some variables in them and notice how the same process works out.0397

What I want to do is simplify the √x6 and I could imagine splitting up my x6 into lots of x2.0404

If I use my rule to split up the root over each of those I will have √x2, √x2, I will have that three different times.0419

Each of these would simplify to just an x.0429

I can go ahead and package that altogether as x3.0433

Notice how that fits with some of our other rules such as rewriting this as x6 / 2 which is x3.0436

All our rule stays nice and consistent with one another, they both agree.0448

Let us try another one.0454

We will slip this one up over 100 and over p8.0457

I could think of the √100 that is 10 and then for the p8 that could be (p4)2.0465

That square and square root can take care of each other 10p4.0475

That one will be good.0481

Let us try another one. This one is √7/y40485

I will put the square root on the top and in the bottom.0492

We want to look at that y4 as (y2)2.0498

In that way we can see that this square and square root will take care of each other.0505

I’m left with the √7/y20511

In this higher root the same rules apply.0523

When it gets down to breaking them up and simplifying them, nothing changes.0526

We just have to be worried about looking for our cubed numbers or in some of these other examples a number raised to the fourth power.0531

Starting with 108, I want to think of this one, I think about a cubed number.0539

108 is the same as 4 × 27.0553

That is important because 27 is one of my cubic numbers.0558

I will split it up over the 4 and 27.0565

We can go ahead and simplify that 3rd root of 27.0570

That one will be just 3.0577

We like to put our numbers first, let me write this one as 3 × 3rd root of 40581

Note how we do not simplify the 4 because it is not a cubic number, it is a square number.0589

Let us try the next one, the 4th root of 160.0597

I want to think of numbers raised to the fourth power.0603

If you want you can even make a list.0606

14 = 1, 24 = 16, this is 4 16 × 10, the 4th root of 16, 4th root of 10.0607

We will go ahead and simplify the 4th root of 16 is 2, the 4th root of 100627

That one is done.0633

Let us use the quotient rule on this last problem.0640

On top I will have the 4th root of 16 and in the bottom 4th root of 6250644

We saw earlier that the 4th root of 16 we will go ahead and simplify that will be just 2.0652

With 625 that is 54, 2/5.0658

Let us do the same thing with higher roots but we will deal with some variables underneath the roots.0671

The first one I have 3rd root of z9.0679

I could look at this as (z3)3.0685

What I want to do in this is highlight this root and that 3rd root end up getting rid of each other.0693

What is left over is z3.0700

Note that this meshes with some of our earlier work and I could write this as z9/3 and I will still get z3.0705

All of our rules are staying nice and consistent with one another.0715

In here the3rd root of 8x6.0719

I’m looking at 3rd root of 8 and 3rd root of x6.0724

That will reduce to 2 and this would be like (x2)3.0730

2 × x2 that will be the final reduced expression.0745

I have the 3rd root of 54 t50753

In this one I will try to break down as much as possible but remember if we have things that are still not cubic,0757

we have to leave them underneath the root.0763

Let us see.0766

What cubic numbers can I find in 54?0767

That will be the same as 27 ×2 and what cubic numbers can I grab from t5?0772

That is a t3 and t2.0785

Notice here I am thinking of my product rule for exponents and how I have to add those exponents together to get 5.0790

I’m splitting them up just this way so that I would have one of them as cubed.0797

We could take the cubed of everything in here.0805

Cube of 27, 2 and t3, t20808

Some of these will simplify and some of them would not.0817

3rd root of 27 is 3, 3rd root of 2 has to stay, 3rd root of t3 is t, and the 3rd root of t2 has to stay.0821

Gathering up what was able to be taken out I will have 3t × 3rd root of 2t20837

Simplify and bring out as much as you can, and if you cannot go ahead and leave them under the root.0846

One more, I will first split up the root over the top and on the bottom.0853

A15 / 3rd root of 640860

On the top of this that is (a5)3 underneath the cube root.0868

For the 64 on the bottom that is something that I could have just take the cube root of.0879

This would simplify into a5 and 3rd root of 64 is 4.0888

That one is simplified.0895

Be familiar with your rules for these radicals especially when it gets to being able to split things up0900

and simplify over each of the individual components.0907

Thank you for watching