42.6 Scientific Notation Multiplication and Division

Scientific Notation Multiplication and Division describes how two numbers written in scientific notation are combined under multiplication or division by operating on their coefficients and their powers of ten separately, then normalizing the resulting expression back into standard scientific form.

Scientific Coefficient Multiplication

Procedure

When multiplying two numbers in scientific notation, their coefficients are multiplied together directly, using ordinary decimal multiplication.

Example

For $2.0·{10}^3$ and $3.0·{10}^4$, the coefficients are multiplied:

Product Exponent Addition

Procedure

The exponents of the two powers of ten are added together, following the product rule for exponents with a common base.

Example

Continuing the same multiplication, the exponents are added:

Scientific Coefficient Division

Procedure

When dividing two numbers in scientific notation, the coefficient of the first number is divided by the coefficient of the second, using ordinary decimal division.

Example

For $8.0·{10}^6$ divided by $2.0·{10}^2$, the coefficients are divided:

Quotient Exponent Subtraction

Procedure

The exponent of the divisor's power of ten is subtracted from the exponent of the dividend's power of ten, following the quotient rule for exponents with a common base.

Example

Continuing the same division, the exponents are subtracted:

Intermediate Scientific Product

Procedure

The multiplied coefficient and the summed exponent are recombined into a single intermediate expression, which may or may not already be in normalized scientific form.

Example

Combining the results from the multiplication example above:

Intermediate Scientific Quotient

Procedure

The divided coefficient and the subtracted exponent are recombined into a single intermediate expression, following the same recombination logic used for a product.

Example

Combining the results from the division example above:

Operation Result Normalization

Procedure

If the resulting coefficient from a multiplication or division falls outside the required range for normalized scientific form, either at or above ten or below one, the expression is renormalized by shifting the decimal point and adjusting the exponent accordingly.

Example

If a multiplication produced a coefficient of $15$ alongside an exponent of $7$, the result $15·{10}^7$ is renormalized to $1.5·{10}^8$.

Scientific Division Nonzero-Divisor Requirement

Requirement

The number being divided by, including both its coefficient and its power of ten, must be nonzero, since division by zero remains undefined regardless of whether the numbers involved are expressed in scientific notation.

Consequence

Since a normalized scientific notation coefficient is already required to have an absolute value of at least one, this requirement is automatically satisfied for any properly normalized divisor, as such a coefficient can never equal zero.