The Zero Exponent Rule
The Zero Exponent Rule states that any non-zero number raised to the power of zero equals 1: a to the 0 = 1 for any a ≠ 0. So 5 to the 0 = 1, (-7) to the 0 = 1, and (1/3) to the 0 = 1. This Grade 7 math skill from Saxon Math, Course 2 follows logically from the pattern of decreasing exponents (4 to the 3 = 64, 4 to the 2 = 16, 4 to the 1 = 4, 4 to the 0 = 1) and is essential for working with scientific notation, polynomial expressions, and all algebraic work involving exponents.
Key Concepts
For any non zero number $a$, a number raised to the power of zero is equal to 1. $$a^0 = 1 \quad (\text{for } a \neq 0)$$.
Common Questions
What is the Zero Exponent Rule?
Any non-zero number raised to the power of zero equals 1. So 7 to the 0 = 1, x to the 0 = 1 (for x ≠ 0), and (3/4) to the 0 = 1.
Why does any number to the zero power equal 1?
The pattern of exponents: each time you decrease the exponent by 1, you divide by the base. 4 to the 3 = 64, 4 to the 2 = 16, 4 to the 1 = 4, 4 to the 0 = 4 divided by 4 = 1.
What is 0 to the 0?
0 to the 0 is considered undefined or indeterminate. The zero exponent rule is only defined for non-zero bases (a ≠ 0).
Does the Zero Exponent Rule apply to negative bases?
Yes. (-5) to the 0 = 1 because the rule applies to any non-zero number, including negatives.
When do students learn the Zero Exponent Rule?
The Zero Exponent Rule is introduced in Grade 7-8. Saxon Math, Course 2 covers it in Chapter 9 alongside negative exponents.
How does the Zero Exponent Rule connect to scientific notation?
In scientific notation, understanding all integer exponents including 0 (any number to the 0 is 1) ensures correct interpretation of expressions like 1.5 times 10 to the 0 = 1.5.
How does the pattern of exponents justify the zero exponent rule?
Divide successive powers by the base: a cubed / a = a squared; a squared / a = a to the 1; a to the 1 / a = a to the 0 = 1. The pattern demands a to the 0 = 1 to remain consistent.