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mathtard

  • 3 years ago

Rewrite with rational exponents. Can someone please help me understand and work through this problem? \[\sqrt[6]{xy^5z}\]

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  1. polpak
    • 3 years ago
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    Ok well do they want you to re-write it with fractional exponents? or ?

  2. mathtard
    • 3 years ago
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    It says re-write with rational exponents

  3. simplephysics
    • 3 years ago
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    Okay, well sqrt6 is like a power of 1/6. Let's see if we can apply that..

  4. myininaya
    • 3 years ago
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    \[\sqrt[a]{x}=x^\frac{1}{a}\]

  5. polpak
    • 3 years ago
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    Ok. So here's basically what you need to know to do all of these problems: \[\huge \sqrt[a]{b^kc^j} = b^{\frac{k}{a}}c^{\frac{j}{a}}\]

  6. myininaya
    • 3 years ago
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    \[\sqrt[a]{b^kc^n}=(b^k c^n)^\frac{1}{a}=b^\frac{k}{a}c^\frac{n}{a}\]

  7. mathtard
    • 3 years ago
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    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?

  8. polpak
    • 3 years ago
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    The index of your radical what you will divide your powers by.

  9. polpak
    • 3 years ago
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    There's an 'is' missing from that sentence.

  10. mathtard
    • 3 years ago
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    I am still confused. I don't know where to begin. I feel like all of these problems are SOOO different

  11. polpak
    • 3 years ago
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    They are not. Remember the last problem you did? \[\sqrt[4]{(5a)^4} = (5a)^{\frac{4}{4}} = (5a)^1 = 5a\]

  12. mathtard
    • 3 years ago
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    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

  13. polpak
    • 3 years ago
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    So in this case we have: \[\sqrt[6]{xy^5z}\] We can rewrite it as: \[\large (xy^5z)^{\frac{1}{6}}\]

  14. polpak
    • 3 years ago
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    And the recall that when you raise a power to a power you multiply the exponents.

  15. polpak
    • 3 years ago
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    Raise a product to a power that is.

  16. polpak
    • 3 years ago
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    You just divide each of the exponents by the index of the radical.

  17. polpak
    • 3 years ago
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    What is the exponent on the x ?

  18. mathtard
    • 3 years ago
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    1?

  19. phi
    • 3 years ago
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    Let's start at the beginning. Do you know how to write \[\sqrt{x}= x^{?}\]

  20. polpak
    • 3 years ago
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    Correct. So now divide that 1 by 6 and the new exponent on the x will be 1/6

  21. mathtard
    • 3 years ago
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    thank you.

  22. polpak
    • 3 years ago
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    Now do the same thing for the y. The exponent on the y is 5, so 5 divided by 6 is 5/6

  23. polpak
    • 3 years ago
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    Then again for the z and your result is: \[\large x^{\frac{1}{6}}y^{\frac{5}{6}}z^{\frac{1}{6}}\]

  24. polpak
    • 3 years ago
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    which doesn't seem 'simpler' at all. But that's sometimes how that goes.

  25. myininaya
    • 3 years ago
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    another general example of applying law of exponent: \[(x^nyz^m)^{r}=(x^ny^1z^m)^r=x^{nr}y^{1r}z^{mr}=x^{nr}y^rz^{mr}\]

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