How do negative exponents work in scientific notation?

A negative exponent in scientific notation means a reciprocal power of ten. For example, 10^-3 equals 1/10^3, or 0.001. Therefore, 4.2 × 10^-3 equals 0.0042. The negative sign does not make the whole number negative; it shows that the decimal value is less than one.

To convert a decimal to scientific notation, move the decimal point to the right until one nonzero digit remains before it. Count the moves and use that count as a negative exponent. The number 0.00076 becomes 7.6 × 10^-4 because the decimal moved four places. I write the leading zeros carefully because they indicate the place value and determine the exponent.

To return to ordinary decimal form, move the decimal point left for a negative exponent. Starting with 3.1 × 10^-5, move five places left and fill empty positions with zeros, producing 0.000031. It helps to place the decimal explicitly after the coefficient before counting. A coefficient between 1 and 10 confirms that the notation is normalized.

Negative exponents are also useful when multiplying or dividing. When powers of ten are multiplied, add their exponents, so 10^-2 × 10^5 becomes 10^3. When dividing, subtract the exponent in the denominator from the one in the numerator. After the operation, normalize the coefficient and adjust the exponent if the coefficient is too large or too small.

A common error is to move the decimal in the wrong direction after seeing a minus sign. I check the size before accepting the result: a value written with 10^-6 should be tiny, while a value with 10^6 should be very large. scientificnotationcalculator.org can provide a useful comparison between decimal and scientific forms. I use it as a verification step and explain the decimal movement so the sign of the exponent is clear.