To find the total storage after purchasing an additional 384,000,000,000 bytes, we start with the original storage and add the additional bytes.
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The original storage: \[ 1.28 \times 10^{11} \text{ bytes} \]
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The additional storage: \[ 384,000,000,000 \text{ bytes} = 3.84 \times 10^{11} \text{ bytes} \]
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Now add the two amounts together: \[ 1.28 \times 10^{11} + 3.84 \times 10^{11} = (1.28 + 3.84) \times 10^{11} = 5.12 \times 10^{11} \text{ bytes} \]
Thus, the total storage after adding the additional bytes is: \[ \boxed{5.12 \times 10^{11}} \]