Multiplicative Inverse Property and Negative Exponents
The multiplicative inverse property and negative exponents is a Grade 7 algebra concept in Big Ideas Math Advanced 2, Chapter 10: Exponents and Scientific Notation. A number raised to a negative exponent is the reciprocal of the same base raised to a positive exponent, because a^n times a^(-n) equals a^0 which equals 1. For example, 3^4 and 3^(-4) are multiplicative inverses since their product equals 1.
Key Concepts
The multiplicative inverse (reciprocal) of $a^n$ is $a^{ n}$, and vice versa: $a^n \cdot a^{ n} = a^0 = 1$.
$$a^n \cdot a^{ n} = 1 \text{ where } a \neq 0$$.
Common Questions
What is a negative exponent?
A negative exponent means taking the reciprocal of the base raised to the positive exponent. For example, a^(-n) equals 1 divided by a^n, so 2^(-3) equals 1 divided by 8.
What is the multiplicative inverse of a^n?
The multiplicative inverse of a^n is a^(-n), because when multiplied together they equal a^0 which equals 1. They are reciprocals of each other.
Why do negative exponents represent reciprocals?
Using the product of powers rule, a^n times a^(-n) equals a^(n plus negative n) equals a^0 equals 1. Since their product is 1, they are multiplicative inverses, which means negative exponents represent reciprocals.
What textbook covers negative exponents in Grade 7?
Big Ideas Math Advanced 2, Chapter 10: Exponents and Scientific Notation covers negative exponents and the multiplicative inverse property.