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Math Help - Irrational number between these two?

  1. #1
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    Irrational number between these two?

    Hi,
    I have two questions:

    1. Is there an irrational number between 2.6 and 2.6repeating (2.666666...)?

    2. Is there an irrational number between 4th root of 2 and fifth root of 2?
    How about 9/2 root of two?

    Thanks for any help!!
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  2. #2
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    Hello, nippy!

    1. Is there an irrational number between 2.6 and 2.666\hdots ?
    Here's a way to solve this . . .

    We have: . 2.6 \:< \:x \:<\: 2.666\hdots

    Square: . 6.76 \:<\:x^2 \:<\: 7.111\hdots

    We can let x^2 equal any number in that interval.

    The simplest is: . x^2 \:=\:7\quad\Rightarrow\quad x \:=\:\sqrt{7}



    2. Is there an irrational number between \sqrt[5]{2} \text{ and }\sqrt[4]{2} ?
    How about 9/2 root of two? . . . . . Yes!
    By my method, we have: . 2^{\frac{1}{5}} \:<\:x\:<\:2^{\frac{1}{4}}

    Raise to the 20th power: . 2^4 \:<\:x^{20} \:<\:2^5\quad\Rightarrow\quad 16 \:<\:x^{20} \:<\:32

    And we can use: . x^{20} \:=\:24\quad\Rightarrow\quad x \:=\:\sqrt[20]{24}

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