Final answer:
In mathematics, specific rules govern operations with exponents. For multiplication, add the exponents; for division, subtract the exponents; for raising a power to another power, multiply the exponents. These rules also apply to scientific notation where coefficients are multiplied or divided, and exponents are added or subtracted accordingly.
Step-by-step explanation:
When dealing with exponents in mathematical expressions, there are specific rules that apply to operations such as multiplication, division, and raising powers to other powers. These rules are fundamental to correctly simplifying expressions and working with numbers, especially in scientific notation.
Raising a Power to Another Power:
To raise an exponential expression to another power, you multiply the exponents. For instance, (a^m)^n = a^(m*n).
Multiplication of Exponentials:
When multiplying two exponential terms with the same base, you keep the base and add the exponents. For example, a^m * a^n = a^(m+n).
Division of Exponentials:
In division, if you have the same base, you subtract the exponent of the divisor from the exponent of the dividend. For example, a^m / a^n = a^(m-n). This rule is akin to what is used in scientific notation, where you divide the coefficients (denoted as N) and subtract the exponents (denoted as n).
Scientific Notation:
In scientific notation, multiplication and division follow similar rules. When multiplying, you multiply the leading coefficients and add the exponents of the powers of ten. Conversely, for division, you divide the leading coefficients and subtract the exponents. This simplifies handling very large or very small numbers, as they can be clearly represented and easily manipulated in this form.
These rules are vital for working with exponentials and scientific notation, and help maintain the order and clarity of numerical expressions.