how can I change this answer into a sig fig answer is 7.800 x 10^22 correct?

.13*6.022 = .78286
just ignore the 10^23

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7.8286x10^22
Alter this to
.78286 x 10^23
just ignore the 10^23 as above.
you are then left with
.78286 as above.

The two equations are equal.

Sig fig means 'Significant figures' Depending on the situation, you may be asked to give your answers in different number of sig fig.

If you are asked to give your answer to 1 sig fig, then, only one 'number' should appear.

Examples of 1 sig fig numbers:
1000
500
20
0.2
0.004

NOT 1 sig fig numbers:
0.0010
1001
200.00

Do you understand now?

The important thing here is that the data that you were given is good to only two significant digits (0.13). Hence the answer that you give is also good to only two significant digits: 7.8 x 10^22. Do NOT put the extra 0's on, as that implies that your result is good to more significant digits than it really is. Hence the answer of 7.8 x 10^22 is "more accurate" than an answer of 7.800 x 10^22.