The  laptop power cable usually consists of two wires or of two wires and the external shield. Sometimes it is made of two wires just insulated in separate insulation and then both insulated together with a one more insulation layer.
Firstly, it is necessary to cut the wires that they both would be of the same length. So as it would be easier to make a good joint. It can be done with any tools that are at home. Here is how the cut wires look.

It is better to use a hot soldering iron for removing the wire insulation. It is a standard procedure in electronics.

The insulation also can be removed with a knife or some other tool. Just remove the insulation equally in length from all of the wires. So, here we have two wires with the insulation removed.

Now we are going to join two wires. The wires are different, which means that one carries the positive voltage and the other - negative voltage. Usually, the wire with the negative voltage is marked with minuses. It is very important to connect two right wires, because, otherwise, we may burn the protection circuit in the laptop motherboard.
So, we are joining two wires like it  is shown in the image below.

Now, when they are stranded together we can solder the joint. (If you do not have a soldering iron just leave it like this, but make sure it is tight and will not unstring easily.)

I have soldered the joint and I will put some isolation tape on it. It is the special insulating tape from the electronics shop. The clear tape is not good for this purpose.

The first joint is finished. Now I will do the same with the second wire.

It is quite easy. Now, I will put some insulating tape on this joint and cover the entire joint:

The result:

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