Operations on Radical Expressions

[GMATJenn]

Hello Everyone....I have a question and the solution...but I don't quite understand it! If anyone knows how to explain it, or has resources that relate to this question, I would be very thankful!

Q. What is the Value of

Thanks for your help everyone!!

 

[Mrspi]

Hello Everyone....I have a question and the solution...but I don't quite understand it! If anyone knows how to explain it, or has resources that relate to this question, I would be very thankful!

Q. What is the Value of

Thanks for your help everyone!!

I can't quite see ALL of your image.....

But, I have an alternate approach to this problem....

I'll use fractional exponents to represent the problem, and try to explain how the rules for exponents lead to the solution. Remember that an exponent of 1/n indicates the nth root. So, sqrt(x) would be x[sup:2d7ldu2l](1/2)[/sup:2d7ldu2l] using a fractional exponent.

Your problem, using fractional exponents, is this:

[(0.000064)[sup:2d7ldu2l](1/3)[/sup:2d7ldu2l]][sup:2d7ldu2l](1/2)[/sup:2d7ldu2l]

One of the rules for exponents says that when you raise a power to a power, you MULTIPLY the exponents.

Symbolically, (a[sup:2d7ldu2l]m[/sup:2d7ldu2l])[sup:2d7ldu2l]n[/sup:2d7ldu2l] = a[sup:2d7ldu2l]m*n[/sup:2d7ldu2l]

Let's apply that rule:

[(0.000064)[sup:2d7ldu2l](1/3)[/sup:2d7ldu2l]][sup:2d7ldu2l](1/2)[/sup:2d7ldu2l] = [0.000064][sup:2d7ldu2l](1/3)*(1/2)[/sup:2d7ldu2l]

or, [0.000064][sup:2d7ldu2l](1/6)[/sup:2d7ldu2l]

Can we write 0.000064 in exponential form? Let's see....that would be (64 / 1000000)

But 64 is 2[sup:2d7ldu2l]6[/sup:2d7ldu2l] and 1 000 000 is 10[sup:2d7ldu2l]6[/sup:2d7ldu2l]............

So, we have

(2[sup:2d7ldu2l]6[/sup:2d7ldu2l] / 10[sup:2d7ldu2l]6[/sup:2d7ldu2l])[sup:2d7ldu2l](1/6)[/sup:2d7ldu2l]

Another of the rules for exponents says that when you raise a fraction to a power, both the numerator and the denominator are raised to that power. Symbolically, (a/b)[sup:2d7ldu2l]m[/sup:2d7ldu2l] = a[sup:2d7ldu2l]m[/sup:2d7ldu2l] / b[sup:2d7ldu2l]m[/sup:2d7ldu2l]

So, [2[sup:2d7ldu2l]6[/sup:2d7ldu2l] / 10[sup:2d7ldu2l]6[/sup:2d7ldu2l]][sup:2d7ldu2l](1/6)[/sup:2d7ldu2l] = [2[sup:2d7ldu2l]6[/sup:2d7ldu2l]][sup:2d7ldu2l](1/6)[/sup:2d7ldu2l] / [10[sup:2d7ldu2l]6[/sup:2d7ldu2l]][sup:2d7ldu2l](1/6)[/sup:2d7ldu2l]

When you multiply the exponents (because you are raising a power to a power), you'll have this:

2/10

But what is that, in decimal form? It is 0.2.

 

[GMATJenn]

Oh ok...that's a different way than explained...But that works too...! I will write it down and practice that strategy, it seems very logical. Thanks! I'm not sure why the image isnt working :s hmmm...

In the image, it showed the solution go through 3 SquareRoot 10^-6 = 10^-2 - what is the rule there? Do you divide the square root by the exponent? This looks simple...but I just dont get it

Thanks for your help Math Experts! I really appreciate your help!

 

[Mrspi]

Oh ok...that's a different way than explained...But that works too...! I will write it down and practice that strategy, it seems very logical. Thanks! I'm not sure why the image isnt working :s hmmm...

In the image, it showed the solution go through 3 SquareRoot 10^-6 = 10^-2 - what is the rule there? Do you divide the square root by the exponent? This looks simple...but I just dont get it

Thanks for your help Math Experts! I really appreciate your help!

"3 SquareRoot" apparently means "cube root".....

We can use a fractional exponent to represent a "cube root"..... the cube root of 2 would be written as (2)[sup:39bozy97](1/3)[/sup:39bozy97]

"cube root of 10[sup:39bozy97]-6[/sup:39bozy97]" is the same thing as (10[sup:39bozy97]-6[/sup:39bozy97]][sup:39bozy97](1/3)[/sup:39bozy97]

I mentioned the rule for raising a power to a power....MULTIPLY the exponents. See how that applies here?

(10[sup:39bozy97]-6[/sup:39bozy97])[sup:39bozy97](1/3)[/sup:39bozy97] = 10[sup:39bozy97](-6)*(1/3)[/sup:39bozy97]

Now....multiply the exponents. But do you remember that multiplying by 1/3 is the same as dividing by 3? So yes, you're right....the exponent "inside" the radical is divided by the "index of the radical"

 

[GMATJenn]

OHHH ok!!! thanks soooo much!!! I get it! So simple...but thanks!!! Thanks you!!!