Laws of Exponents 

The rules for simplifying with exponents are as follows:

  • Product property: ( xm ) ( xn ) = xm + n
  • Power of a power property: ( xm )n = xm × n
  • Power of a product property: (xy)m = xmym
  • Power of a quotient property: (xy)m=xmym
  • Quotient property: xmxn=xm−n
  • Zero power property: x0 = 1
  • Negative power property: The rule is expressed a negative exponent inverts the base, making the exponent positive. See the attached link for more detail - https://youtu.be/Q74fghBPbWE?si=onFLFqp4AzBy-PYJ   

I like to break the rules or theorems of Exponents down into three distinct categories : Core Rules, Special Exponents, and Grouped Expressions. 

Core rules

Which consist of:

A. Product of powers -When multiplying two exponents with the same base, add the exponents

B. Quotient of powers -When dividing two exponents with the same base, subtract the exponent in the denominator from the exponent in the numerator.

C. Power of a power- When raising a power to another power, multiply the exponents.

 

Special Exponents 

Which consist of :

A. Zero exponent- Any non-zero number raised to the power of zero is equal to one.

B. Negative exponent- A negative exponent indicates the reciprocal of the base raised to the positive exponent.

 

Grouped Expressions

Which consist of:

A. Power of a product-To raise a product to a power, raise each factor in the product to that power.

B. Power of a quotient- To raise a quotient to a power, raise both the numerator and the denominator to that power.

     

    More facts about Exponents 

    Real-World Applications

    Scientific Notation: Exponents, particularly powers of 10, are essential for writing very large or very small numbers in a compact form

     

    • Measurement: Exponents are used in calculating area (squared, e.g., square feet) and volume (cubed, e.g., cubic feet).

     

    • Growth and Decay: They describe processes with rapid growth, such as population growth, compound interest, or bacterial multiplication. 
    • (Access the link below for more information on exponents)

    https://science.howstuffworks.com/math-concepts/exponent-rules.htm