Rules of Exponents

Definitions

Terminology
We would call this “x raised to the a power”
Rules_of_exponents_1.png
x is called the base
a is called the exponent

Example:
Rules_of_exponents_2.png is called “seven raised to the eleventh power”
the base is 7
the exponent is 11

Integer exponents
An integer exponent just means multiplying a number by itself.
Rules_of_exponents_3.png

Example:
Rules_of_exponents_4.png
Rules_of_exponents_5.png

Fraction exponents
An exponent which is the reciprocal of an integer means taking the root
Rules_of_exponents_6.png

Example:
Rules_of_exponents_7.png
Rules_of_exponents_8.png

Negative exponents
A negative exponent indicates a reciprocal
Rules_of_exponents_9.png

Example:
Rules_of_exponents_10.png
Rules_of_exponents_11.png

NOTICE: This means you don’t ever have to deal with fractions or roots ever again if you don’t want to! You can treat them all like exponents.

Algebra rules

Combining exponents
When two numbers with the same base are multiplied, the exponents may be added
Rules_of_exponents_12.png

Example:
Rules_of_exponents_13.png
Rules_of_exponents_14.png

Exponents in ratios
When exponents in the numerator an denominator have the same base, their exponents may be subracted
Rules_of_exponents_15.png

NOTE: This is just a combination of the negative exponent rule with the multiplication rule!

Distributing exponents
If you have 2 layers of exponents, they may be combined
Rules_of_exponents_16.png

Example:
Rules_of_exponents_17.png
Rules_of_exponents_18.png

Negative bases
When the base is a negative number, the sign of the expression overall depends on whether the exponent is even or odd.
Rules_of_exponents_19.png   if n is an odd integer
Rules_of_exponents_20.png   if n is an even integer

Therefore:
Rules_of_exponents_21.png   if n is an odd integer
Rules_of_exponents_22.png   if n is an even integer

Examples:
Rules_of_exponents_23.png
Rules_of_exponents_24.png

A warning about addition
Exponents DO NOT distribute cleanly over addition
Rules_of_exponents_25.png

You either have to find a way to simplify the base first OR use the definition of the exponent to simplify things.
Examples:
Rules_of_exponents_26.png

Rules_of_exponents_27.png

Rules_of_exponents_28.png
Rules_of_exponents_29.png

Author: Wade

Materials scientist, lab manager, and educator based in the Seattle area.

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