It’s not an exact meaning to “multiplying by itself”
Clearly you can’t multiply 10 by itself by point whatever times.
But you can fill in the gaps with a different math function. (In this case exponentiation). As long as the exponent is a whole number it’s the same as multiplying the number by itself. But if it’s a decimal, it just isn’t the same as repeated multiplication and can only be understood as exponentiation.
So yeah, the answer is 10^0.778
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