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# Rewriting roots as rational exponents

Sal solves several problems about the equivalence of expressions with roots and rational exponents. For example, rewrite ⁶√(g⁵) as g^⅚.

## Want to join the conversation?

- When am I going to use this math in life? This is a serious question.(27 votes)
- Well, what profession are you considering?

Even if you're interested in something like painting, where math isn't that involved, an understanding of exponents is vastly helpful in managing money, savings, interest rates, investments, etc., which everyone certainly has to deal with at some point. Hope that I helped.(42 votes)

- because im english i dont understand what a constant is please help :)(16 votes)
- A constant is a number which doesn't change, unlike a variable, which can change, i.e. d at3:00has only one value, while x can have any value greater than zero for the equation to be true.(8 votes)

- Someone give me an example of how I'm gonna use in life...(2 votes)
- Howdy EhP,

It really depends how creative and curious a mind you are. People often complain about learning math and "how will it be useful". They say that since they will never be an engineer or scientists, they shouldn't have to learn it.

But instead of thinking of it that way, think of the opportunities you could give yourself. When I was young, I put in the hard work to learn Algebra, and now I use it all over the place. Whether it is programming, computer science, physics, chemistry, engineering, etc, the skills learned in Algebra can come in very handy. Now it is true that not all of these tools will help me make money--at least not directly. However, as I become smarter I will be able to take what I have learned elsewhere and apply it to other skills. In fact, many times great things are discovered when you take what you learn from one skillset and apply it to another skillset.

So my advice to you: think of every skill you learn as an opportunity, not a chore. Eventually, you will begin to see Algebra in places that you wouldn't have if you weren't looking for it.

Learn on, my friend.(19 votes)

- At about 0.50, why do you multiply the two numbers for v to the 3rd to the 1/7? I kind of get it, but I'm still a little confused. Wouldn't it be v to the 3 1/7?(4 votes)
- The properties of exponents specify that when one exponent is raised to another exponent, you multiply the exponents. for example: (x^2)^3 = x^(2*3) = x^6.

So, in the video, Sal has (v^3)^(1/7). Multiply the exponents: v^(3/1 * 1/7) = v^(3/7)

Hope this helps.(3 votes)

- how do you rewrite the root if its a negitive decimal with a fraction as an exponent?(3 votes)
- FIrst off, we cannot have a negative on any of our even roots (square, 4th, 6th, etc.) without getting into imaginary numbers. So if you are asking (-.5)^(1/5) we could write the square root sign with a raised 5 on the crook of the root sign, and a -.5 inside.

Hope this helps.(5 votes)

- How does3:30work, when x has a positive exponent, turn into a negative?

I understand that x^-2 is the same as 1/x... though I dont understand how that works either, it's just memorised.(3 votes)- Actually,
`x^-2`

is the same as`1/x^2`

. Sal has a great set of videos on negative exponents, including a video explaining why negative exponents work: https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-numbers-operations/cc-8th-pos-neg-exponents/v/negative-exponents(3 votes)

- But what if there is more than one square root on the base? For example, log a^1/2^a^2.(3 votes)
- Ellis helped me with these problems,now i teach a harverd school.(3 votes)
- I understand this and all but what is a constant when it says d is an constant(1 vote)
- That means d represents a single value, as opposed to, being related to another variable (like p=3d, which could have many(infinite) d values and many(infinite) corresponding p values.(4 votes)

- Sometimes it says: "The equation is true for x>0." Why is this important?(0 votes)
- You cannot take the even roots of negative numbers. That means when an exponent is in rational form, if the denominator is even, then you have to say that x≥0. However, if the denominator is odd, the domain is all real numbers.(5 votes)

## Video transcript

- [Voiceover] We're asked to determine whether each expression is equivalent to the seventh root of v to the third power. And, like always, pause the
video and see if you can figure out which of
these are equivalent to the seventh root of v to the third power. Well, a good way to figure
out if things are equivalent is to just try to get
them all in the same form. So, the seventh root of
v to the third power, v to the third power, the seventh root of
something is the same thing as raising it to the 1/7 power. So, this is equivalent
to v to the third power, raised to the 1/7 power. And if I raise something to
an exponent and then raise that to an exponent, well
then, that's the same thing as raising it to the product
of these two exponents. So, this is going to be the same thing as v to the three times 1/7 power, which, of course, is 3/7. 3/7. So, we've written it
in multiple forms now. Let's see which of these match. So, v to the third to the 1/7 power, well, that was the form that
we have right over here, so that is equivalent. V to the 3/7. That's what we have right over here, so that one is definitely equivalent. Now, let's think about this one. This is the cube root of v to the seventh. Is this going to be equivalent? Well, one way to think about it, this is going to be the same thing as v to the 1/3 power ... actually, no, this wasn't the
cube root of v to the seventh, this was the cube root of v,
and that to the seventh power. So, that's the same thing
as v to the 1/3 power, and then, that to the seventh power. So, that is the same thing
as v to the 7/3 power, which is clearly different
to v to the 3/7 power. So, this is not going to
be equivalent for all v's, all v's for which this
expression is defined. Let's do a few more of these,
or similar types of problems dealing with roots and
fractional exponents. The following equation is true for g greater than or equal to
zero, and d is a constant. What is the value of d? Well, if I'm taking the
sixth root of something, that's the same thing as
raising it to the 1/6 power. So, the sixth root of g to the fifth, is the same thing as g to the fifth, raised to the 1/6 power. And, just like we just
saw in the last example, that's the same thing as g
to the five times 1/6 power. This is just our exponent properties. I raise something to an exponent and then raise that whole thing
to another exponent, I can just multiply the exponents. So, that's the same thing as g to the 5/6 power. And so d is 5/6. Five over six. The sixth root of g to the
fifth is the same thing as g to the 5/6 power. Let's do one more of these. The following equation is
true for x greater than zero, and d is a constant. What is the value of d? Alright, this is interesting. And I forgot to tell
you in the last one, but pause this video as
well and see if you can work it out on ...or pause
for this question as well and see if you can work it out. Well, here, let's just start rewriting the root as an exponent. So, I can rewrite the whole thing. This is the same thing as one over, instead of writing the seventh root of x, I'll write x to the 1/7 power is equal to x to the d. And if I have one over
something to a power, that's the same thing as that something raised to the negative of that power. So, that is the same thing as x to the negative 1/7 power. And so, that is going to
be equal to x to the d. And so, d must be equal to, d must be equal to negative 1/7. So, the key here is when
you're taking the reciprocal of something, that's the
same thing as raising it to the negative of that exponent. Another way of thinking about it is you could view this as, you could view it as, x to the 1/7 to the negative one power. And then, if you multiply these exponents, you get what we have right over there. But, either way, d is
equal to negative 1/7.