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mathtard
Rewrite with rational exponents. Can someone please help me understand and work through this problem? \[\sqrt[6]{xy^5z}\]
Ok well do they want you to re-write it with fractional exponents? or ?
It says re-write with rational exponents
Okay, well sqrt6 is like a power of 1/6. Let's see if we can apply that..
\[\sqrt[a]{x}=x^\frac{1}{a}\]
Ok. So here's basically what you need to know to do all of these problems: \[\huge \sqrt[a]{b^kc^j} = b^{\frac{k}{a}}c^{\frac{j}{a}}\]
\[\sqrt[a]{b^kc^n}=(b^k c^n)^\frac{1}{a}=b^\frac{k}{a}c^\frac{n}{a}\]
so how do I start so I take the b and move it over? make a the denominator and make k and j the numerators?
The index of your radical what you will divide your powers by.
There's an 'is' missing from that sentence.
I am still confused. I don't know where to begin. I feel like all of these problems are SOOO different
They are not. Remember the last problem you did? \[\sqrt[4]{(5a)^4} = (5a)^{\frac{4}{4}} = (5a)^1 = 5a\]
It is easy to you because you understand I don't. I just don't learn that way. I need to see how it is solved. I have to actually be able to do the problem
So in this case we have: \[\sqrt[6]{xy^5z}\] We can rewrite it as: \[\large (xy^5z)^{\frac{1}{6}}\]
And the recall that when you raise a power to a power you multiply the exponents.
Raise a product to a power that is.
You just divide each of the exponents by the index of the radical.
What is the exponent on the x ?
Let's start at the beginning. Do you know how to write \[\sqrt{x}= x^{?}\]
Correct. So now divide that 1 by 6 and the new exponent on the x will be 1/6
Now do the same thing for the y. The exponent on the y is 5, so 5 divided by 6 is 5/6
Then again for the z and your result is: \[\large x^{\frac{1}{6}}y^{\frac{5}{6}}z^{\frac{1}{6}}\]
which doesn't seem 'simpler' at all. But that's sometimes how that goes.
another general example of applying law of exponent: \[(x^nyz^m)^{r}=(x^ny^1z^m)^r=x^{nr}y^{1r}z^{mr}=x^{nr}y^rz^{mr}\]