Exponents and Scientific Notation
Lessons in this chapter
About Exponents and Scientific Notation
Exponents and Scientific Notation builds a full set of exponent rules, then uses them to make very large and very small numbers easy to write and compare.
Writing and Evaluating Exponents reviews exponent notation itself before Product and Power Properties and Quotient Properties and Combining Rules build the core rules for multiplying, dividing, and raising powers to a power.
Zero and Negative Exponents extends those same rules to cases that don't involve repeated multiplication in the usual sense, establishing why any nonzero number to the zero power is 1, and why a negative exponent means a reciprocal.
Approximating and Comparing Large and Small Numbers, Writing Numbers in Scientific Notation, and Operations in Scientific Notation close the chapter by applying every exponent rule to real large and small numbers, writing them compactly and calculating with them without writing out long strings of zeros.
All seven lessons are 8th grade math worksheets worked inside the page, so Product and Power Properties, Zero and Negative Exponents and Operations in Scientific Notation are runs of math problems rather than a rules table to memorize. Each answer is marked as it is entered, which is where the common error of adding exponents when the rule calls for multiplying them surfaces. The chapter is free to try, with no login needed for your first 3.
Frequently asked questions
Writing and Evaluating Exponents opens the chapter with base and exponent notation, then Product and Power Properties and Quotient Properties and Combining Rules build the full set of exponent rules. Zero and Negative Exponents explains why anything to the power zero is 1 and what a negative exponent means, and the last three lessons, Approximating and Comparing Large and Small Numbers, Writing Numbers in Scientific Notation and Operations in Scientific Notation, put those rules to work on very large and very small quantities. This sits in the Common Core domain Expressions and Equations (8.EE).
The product property says multiplying powers with the same base means adding the exponents; the power property says raising a power to another power means multiplying the exponents; the quotient property says dividing powers with the same base means subtracting the exponents.
Any nonzero number raised to the zero power equals 1. This follows directly from the quotient property, since dividing a power by itself (like x³ ÷ x³) gives x⁰, and anything divided by itself is 1.
A negative exponent means take the reciprocal of the base raised to the positive version of that exponent: x⁻² equals 1/x², not a negative number.
Scientific notation writes a number as a value between 1 and 10 multiplied by a power of 10: it's used to write very large or very small numbers compactly, without writing out long strings of zeros.
Move the decimal point until only one nonzero digit remains to its left, then count how many places it moved: that count becomes the power of 10 (positive for large numbers, negative for small ones).