Exponential notation and algebraic laws for powers, includingĀ āa0=1,Ā negative exponents, multiplying and dividing powers with the same base, exponents of products and quotients and powers of powersĀ ā(am)n=amn.
Exponential expressions are a shortcut for writing the multiplication of a number by itself many times:
HereĀ āaāĀ is called the base andĀ ānāĀ the exponent. We say thatĀ āaāĀ is raised to theĀ ānthāĀ power.
Note thatĀ āa1=a,Ā since we haveĀ ā1Ća=a.
Any number raised to the power zero is
And since any number multiplied byĀ ā0āĀ isĀ ā0:
WhenĀ ān=0,Ā we haveĀ ā00,Ā which is technically undefined, but in most contexts is defined to be
When multiplying exponentials with the same base, the following rule applies:
An exponential can be the base of another exponential:
In general,
When exponentials with the same power are being multiplied or divided, the bases can be combined:
Nice work completing Exponentiation and integer exponent laws, here's a quick recap of what we covered:
Exercises checked off