What is (1.05×10^23)×6.135 rounded to three sig figs?

The correctly rounded result of one point zero five times ten to the twenty-third multiplied by six point one three five is one point seven one times ten to the twenty-four when rounded to three significant figures.

You multiply the two numbers by treating the ten power part and the decimal part separately, so one point zero five times ten to the twenty-third times six point one three five equals the product of one point zero five and six point one three five times ten to the twenty-third. Multiply one point zero five and six point one three five to get six point four four one seven five, then adjust the power of ten by converting six point four four one seven five times ten to the twenty-third into one point six four four one seven five times ten to the twenty-four so the decimal sits between one and ten. Rounding that value to three significant figures gives one point seven one times ten to the twenty-four because you keep the first three nonzero digits and increase the last kept digit if the next digit is five or greater.

A common mistake is forgetting to adjust the power of ten after multiplying the decimal parts, which can give an answer with the wrong exponent, so always write the decimal in proper scientific notation between one and ten; another error is rounding too early, which loses precision, so do all multiplication first and then round at the end; some students also miscount significant figures, so remember to count nonzero digits and trapped zeros, and round to the required number of significant figures, not decimal places.

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Scientific Notation: Multiply and Round to 3 Sig Figs

1. Multiply decimals and powers separately
2. Compute decimal product first
3. Convert to 1–10 decimal range
4. Adjust power of ten accordingly
5. Round after completing multiplication

Common Scientific Notation Mistakes and Fixes

1. Not adjusting the power of ten
2. Rounding too early and losing precision
3. Miscounting significant figures

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