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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.

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• does that mean that the I can't simplify the cube root of 3?
• 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
• (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..
• 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.
• i dont get how letters can be perfect or not perfect
• 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
• What about bigger roots, like root 4? A problem in my book is: simplify ^4 of z^8. What would I do there?
• 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.
• 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!

Cheers!
• 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.
• what about taking the absolute value here , is it not necessary in case of cube root?
• 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.
• What if the constant was not a perfect cube? Like the number 24? How would you do that? Thanks
• Is there any other way to get the right answer??
• 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!