Your checkNotNegative method doesn't just check. It asserts or requires that it's not negative. I would expect a check to return a Boolean result, not throw an exception.

You can avoid the repeated calls to myPow if you store the result in an array.

final long[] c = new int[a.length + b.length - 1];

        c[i+j] = a[i] * b[j];

long multiplier = 1;
for (long multiplicand : c) {
    product += multiplicand * multiplier;
    multiplier *= 10;
}

This does one more multiplication than the greatest call to myPow but replaces all the calls.

It's trickier, but you could probably do the same thing without the array.

You can avoid the repeated calls to myPow if you store the result in an array.

final long[] c = new int[a.length + b.length - 1];

        c[i+j] = a[i] * b[j];

long multiplier = 1;
for (long multiplicand : c) {
    product += multiplicand * multiplier;
    multiplier *= 10;
}

This does one more multiplication than the greatest call to myPow but replaces all the calls.

It's trickier, but you could probably do the same thing without the array.