you can use akeeba backup component to move your site to anywhere.
Install akeebabackup component
take a back up of your site using component
download your backup
get kickstart.php file from akeeba site.
upload backup and kickstart.php file to location where you want to reinstall the joomla (in your case it will be root public_html)
go to your site
www.mywebsite.com.au/kickstart.php and than follow the instrction(Somewhat similar to joomla installation)
once everything is done you can see the same copy of your backedup website but at new location.
Let me know if you have any troubles.
Cheers,
Nikunj