Irrational number between these two?

**nippy** · January 24th 2008, 07:48 PM

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!!

**Soroban** · January 24th 2008, 08:24 PM

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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