42.8 Integer Powers of Scientific Notation

Integer Powers of Scientific Notation describes how an entire number written in scientific notation is raised to an integer power, by applying that outer exponent separately to the coefficient and to the power of ten, then normalizing the result back into standard scientific form.

Grouped Scientific Number Base

Description

The entire scientific notation expression, consisting of the coefficient multiplied by a power of ten, is treated as a single grouped base when an outer exponent is applied to it.

Purpose of This Grouping

Treating the expression as a single grouped base allows the power-of-a-product rule to be applied, distributing the outer exponent $k$ to both the coefficient and the power of ten separately.

Scientific Coefficient Integer Power

Procedure

The coefficient of the scientific notation expression is raised to the outer exponent using ordinary exponent evaluation.

Example

For ${(2·{10}^3)}^2$, the coefficient is raised to the power:

Power-of-Ten Exponent Multiplication

Procedure

The exponent already attached to the power of ten is multiplied by the outer exponent, following the power-of-a-power rule for exponents.

Example

Continuing the same expression, the exponent on the power of ten is multiplied:

Powered Result Normalization

Procedure

The newly computed coefficient and the newly computed exponent are recombined, and if the resulting coefficient falls outside the range required for normalized scientific form, the expression is renormalized by shifting the decimal point and adjusting the exponent.

Example

Combining the results above produces $4·{10}^6$, which is already within the required coefficient range and needs no further renormalization.

Zero-Power Scientific Expression

Description

When the outer exponent applied to an entire scientific notation expression is zero, the whole expression simplifies directly to one, following the same zero-exponent rule established for any nonzero base.

Example

Negative Integer Power Reciprocal Conversion

Procedure

When the outer exponent applied to an entire scientific notation expression is negative, the reciprocal rule is applied to the whole expression first, and the positive version of the exponent is then distributed to the coefficient and the power of ten as usual.

Example

This reciprocal result can then be rewritten with a negative power of ten, producing $2.5·{10}^{-7}$, restoring normalized scientific form.