The database cannot be opened because it is version 782. This server supports version 706 and earlier. A downgrade path is not supported

Try changing Tools > Options > Database Tools > Data Connections > SQL Server Instance Name.

The default for VS2013 is (LocalDB)\v11.0.

Changing to (LocalDB)\MSSQLLocalDB, for example, seems to work - no more version 782 error.

Try to change the compatibility level, worked for me.

Verify what level it is

USE VJ_DATABASE;
GO
SELECT compatibility_level
FROM sys.databases WHERE name = 'VJ_DATABASE';
GO

Then make it compatible with the older version

ALTER DATABASE VJ_DATABASE
SET COMPATIBILITY_LEVEL = 110;   
GO

• 100 = Sql Server 2008
• 110 = Sql Server 2012
• 120 = Sql Server 2014

By default, Sql Server 2014 will change the db versions compatibility to only 2014, using the @@ version you should be able to tell, which version Sql Server is.

Then run the command above to change it the version you have.

Additional step: Ensure you look at the accessibility of the DB is not reset, do this by right clicking on properties of the folder and the database. (make sure you have rights so you don't get an access denied)

For me using solution provided by codedom did not worked. Here we can only changed compatibility version of exiting database.

But actual problem lies that, internal database version which do not matches due to changes in there storage format.

Check out more details about SQL Server version and their internal db version & Db compatibility level here So it would be good if you create your database using SQL Server 2012 Express version or below. Or start using Visual Studio 2015 Preview.