# Thread: Irrational number between these two?

1. ## 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!!

2. Hello, nippy!

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

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

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

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

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

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

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

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

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### what is between irrational number 2 and 2.6

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