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## Algebra (all content)

### Unit 11: Lesson 8

Radicals (miscellaneous videos)- Simplifying square-root expressions: no variables
- Simplifying square roots of fractions
- Simplifying rational exponent expressions: mixed exponents and radicals
- Simplifying square-root expressions: no variables (advanced)
- Intro to rationalizing the denominator
- Worked example: rationalizing the denominator
- Simplifying radical expressions (addition)
- Simplifying radical expressions (subtraction)
- Simplifying radical expressions: two variables
- Simplifying radical expressions: three variables
- Simplifying hairy expression with fractional exponents

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# Simplifying radical expressions: three variables

A worked example of simplifying the cube root of 27a²b⁵c³ using the properties of exponents. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- does that mean that the I can't simplify the cube root of 3?(17 votes)
- we can not simplify cube root 3...but..we can find its value by method of logaritms

say y=cube root3...taking log on both sides, log y =log(cuberoot 3)=(1/3)(log 3)

=(1/3)*0.48=0.16....now, log y =0.16....now we take antilog on both sides,,,,

y=antilog 0.16 = 1.4454.....so, cuberoot 3=1.4454(2 votes)

- (3^3) X ( b^3) X (c^3) = (3bc)^ 3

which property of exponents is Sal referring to,

is it : (a^m)^n = (a)^ mn...

my memory has misplaced this property.....oh

Also please guide me to the video which Sal is referring to .. i intend to review it..(6 votes)- There's a separate rule that (a^m) * (b^m) = (ab)^m.

If you think about it, it all comes down to basic principles. a^m is just m copies of a multiplied together, right? So a^m b^m is just a*a*...**a*b*b**...**b, with n copies of each, so we can rearrange those terms using the commutative property to get that it is also a*b*a*b**...*a*b, which is n copies of ab multiplied together or (ab)^n.(13 votes)

- i dont get how letters can be perfect or not perfect(3 votes)
- Perfect numbers just mean it has a square root that is a whole number. Non perfect numbers have a square root has is not a whole number and has decimals.

Example:

Perfect:4,9,16,25,36,49,64,81

Non Perfect:3,5,7,45,56,67,78(17 votes)

- What about bigger roots, like root 4? A problem in my book is: simplify ^4 of z^8. What would I do there?(5 votes)
- Roots can be turned into fractional exponents. the n'th root can be re-written as an exponent of 1/n. So the 4th root of x is the same as x^(1/4). Once you get that, then you should be able to use properties of exponents to finish the problem.(5 votes)

- Hi!

I would like to know why the cube root of a given number is equal to that same number to the power of 1/3,

³√x = x to the 1/3 power.

Why 1/3? What makes it 1/3? That's something I'm having a little trouble understanding!

Thanks in advance!

Cheers!(3 votes)- That's a good question. Take a look at this:

We know that when you multiply numbers that have exponents, you add the exponents, right? So for example, 2^3 * 2^2 = 2^5. And likewise, 2^1 * 2^1 *2^1 = 2^3, which equals 8. Now let's try it with a variable for the exponent, where we are trying to find the cube root of 8 by raising 8 to some undetermined power:

8^x * 8^x * 8^x = 8^1 = 8. What does x have to be? When I add up three x's, I have to get 1. 3x =1. x = 1/3.

So 8^(1/3) is the cube root of 8.

You can show the same thing using the rule that says (a^n)^m = a^(n*m)

(8^(1/3))^3 = 8^(1/3*3) = 8^1.(7 votes)

- what about taking the absolute value here , is it not necessary in case of cube root?(2 votes)
- i'm guessing you meant 'do we have to do like square root and take the positive root?'

cube root is pretty much the opposite of taking the number to the power of 3

so...

2^3= 8

cuberoot(8)=2

-2^3 = -2*-2*-2 = -8

cuberoot(-8)= -2

if you get cuberoot(8), you have only 1 answer.(5 votes)

- What if the constant was not a perfect cube? Like the number 24? How would you do that? Thanks(3 votes)
- Is there any other way to get the right answer??(3 votes)
- This is the process and Sal demonstrated it in excruciating detail. As you get used to the procedure, most of it you will be able to do in your head.

Keep practicing!(3 votes)

- Okay, I get this. Now what to do with the cube root of 9+4√5? According to wolfram and my calc it can be simplified to (3+√5)/2. I get the simplification steps wolfram offers, but I couldn't find the general way to simplify this kind of expressions. (Some other examples are the cube root of 40+11√13 which is (5+√13)/2, and the cube root of 10-6√3, which is 1-√3..)(3 votes)
- thank you, cause khan academy and my teacher make me get everything happen in my class . love your(1 vote)

- Would this method work for higher roots, like fourth and fifth roots?(3 votes)
- Yes for example ^4 root of 16=2(1 vote)

## Video transcript

We're asked to simplify the
cube root of 27a squared times b to the fifth times
c to the third power. And the goal, whenever
you try to just simplify a cube root like
this, is we want to look at the parts of
this expression over here that are perfect cubes,
that are something raised to the third power. Then we can take just
the cube root of those, essentially taking them
out of the radical sign, and then leaving
everything else that is not a perfect
cube inside of it. So let's see what we can do. So first of all, 27--
you may or may not already recognize this
as a perfect cube. If you don't already
recognize it, you can actually do
a prime factorization and see it's a perfect cube. 27 is 3 times 9,
and 9 is 3 times 3. So 27-- its prime factorization
is 3 times 3 times 3. So it's the exact same thing
as 3 to the third power. So let's rewrite this
whole expression down here. But let's write it in terms of
things that are perfect cubes and things that aren't. So 27 can be just rewritten
as 3 to the third power. Then you have a squared--
clearly not a perfect cube. a to the third would have been. So we're just going
to write this-- let me write it over here. We can switch the order
here because we just have a bunch of things being
multiplied by each other. So I'll write the a
squared over here. b to the fifth is not a
perfect cube by itself, but it can be expressed
as the product of a perfect cube and
another thing. b to the fifth is the exact same thing as
b to the third power times b to the second power. If you want to see that
explicitly, b to the fifth is b times b times
b times b times b. So the first three are
clearly b to the third power. And then you have b to
the second power after it. So we can rewrite b to
the fifth as the product of a perfect cube. So I'll write b
to the third-- let me do that in that
same purple color. So we have b to the
third power over here. And then it's b to the
third times b squared. So I'll write the b
squared over here. And we're assuming we're going
to multiply all of this stuff. And then finally, we
have-- I'll do in blue-- c to the third power. Clearly, this is a perfect cube. It is c cubed. It is c to the third power. So I'll put it over here. So this is c to the third power. And of course, we still have
that overarching radical sign. So we're still trying to take
the cube root of all of this. And we know from our
exponent properties, or we could say from
our radical properties, that this is the
exact same thing. That taking the cube root
of all of these things is the same as
taking the cube root of these individual factors
and then multiplying them. So this is the same
thing as the cube root-- and I could separate
them out individually. Or I could say the cube
root of 3 to the third b to the third c to the third. Actually, let's do it both ways. So I'll take them
out separately. So this is the same thing
as the cube root of 3 to the third times the cube
root-- I'll write them all in. Let me color-code it so we don't
get confused-- times the cube root of b to the third
times the cube root of c to the third times
the cube root-- and I'll just group
these two guys together just because we're not
going to be able to simplify it any more-- times the cube
root of a squared b squared. I'll keep the colors
consistent while we're trying to figure
out what's what. And I could have said that
this is times the cube root of a squared times
the cube root of b squared, but that won't
simplify anything, so I'll just leave
these like this. And so we can look at
these individually. The cube root of 3 to the
third, or the cube root of 27-- well, that's clearly
just going to be-- I want to do that in
that yellow color-- this is clearly
just going to be 3. 3 to the third power is
3 to the third power, or it's equal to 27. This term right over
here, the cube root of b to the third--
well, that's just b. And the cube root
of c to the third, well, that is
clearly-- I want to do that in that-- that
is clearly just c. So our whole expression has
simplified to 3 times b times c times the cube root
of a squared b squared. And we're done. And I just want to
do one other thing, just because I did mention
that I would do it. We could simplify it this way. Or we could recognize that
this expression right over here can be written as 3bc
to the third power. And if I take three
things to the third power, and I'm multiplying it,
that's the same thing as multiplying them
first and then raising to the third power. It comes straight out of
our exponent properties. And so we can rewrite
this as the cube root of all of this times
the cube root of a squared b squared. And so the cube
root of all of this, of 3bc to the third power, well,
that's just going to be 3bc, and then multiplied by the cube
root of a squared b squared. I didn't take the trouble
to color-code it this time, because we already figured
out one way to solve it. But hopefully, that
also makes sense. We could have done
this either way. But the important thing is
that we get that same answer.