Rational numbers raised to an irrational power.

2d3vljtq
2022-07-09
Answered

Rational numbers raised to an irrational power.

You can still ask an expert for help

tilsjaskak6

Answered 2022-07-10
Author has **14** answers

${2}^{{\mathrm{log}}_{2}3}=3$ is rational.

Check that ${\mathrm{log}}_{2}3$ is irrational.

Suppose it is rational.

${\mathrm{log}}_{2}3=\frac{a}{b}$

where $gcd(a,b)=1$.

${3}^{b}={2}^{a}$which is a contradition.

Check that ${\mathrm{log}}_{2}3$ is irrational.

Suppose it is rational.

${\mathrm{log}}_{2}3=\frac{a}{b}$

where $gcd(a,b)=1$.

${3}^{b}={2}^{a}$which is a contradition.

asked 2021-12-14

What is the square root of 80?

asked 2022-05-18

For an irrational number $\alpha $, prove that the set $\{a+b\alpha :a,b\in \mathbb{Z}\}$ is dense in $\mathbb{R}$.

asked 2022-11-05

On the number line, are there more rational numbers or irrational numbers?

asked 2022-08-07

Write the statement in the form 'if p, then q'

All whole numbers are rational.

Choose the statement that best rewrites the sentence in the specified form.

A. If a number is whole, then it is always rational.

B. If a number is whole, then it is never rational.

C. If a number is rational, then it is never whole.

All whole numbers are rational.

Choose the statement that best rewrites the sentence in the specified form.

A. If a number is whole, then it is always rational.

B. If a number is whole, then it is never rational.

C. If a number is rational, then it is never whole.

asked 2022-06-19

Let $\sqrt{2}-\sqrt{3}$, $\frac{\sqrt{8}}{\sqrt{2}}$. How to find if the result of a problem is a rational or irrational number without solving the problem?

asked 2021-02-21

Which statement is false?

A. every irrational number is also a real number.

B. every integer is also a real number.

C. no irrational number is irrational.

D. every integer is also an irrational number.

A. every irrational number is also a real number.

B. every integer is also a real number.

C. no irrational number is irrational.

D. every integer is also an irrational number.

asked 2022-07-15

Prove that for any distinct primes $p$ and $q$, the ratio $\frac{\sqrt{p}}{\sqrt{q}}$ is irrational.