Exponential notation and algebra, including laws for zero and negative exponents, multiplying and dividing powers with the same base, products and quotients with the same exponent, powers of powers, equating exponents in exponential equations, and rationalizing denominators when needed.
Exponential expressions are a shortcut for writing the multiplication of a number by itself many times:
Here a is called the base and n the exponent. We say that a is raised to the nth power.
Note that a1=a, since we have 1×a=a.
Any number raised to the power zero is
And since any number multiplied by 0 is 0:
When n=0, we have 00, 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:
If two exponentials in the same positive base are equal, their exponents must be equal:
Exponentials can also appear in equations with one or more unknown:
Now we can equate the exponents:
Nice work completing Exponentiation and integer exponent laws, here's a quick recap of what we covered:
Exercises checked off
Exponential notation and algebra, including laws for zero and negative exponents, multiplying and dividing powers with the same base, products and quotients with the same exponent, powers of powers, equating exponents in exponential equations, and rationalizing denominators when needed.
Exponential expressions are a shortcut for writing the multiplication of a number by itself many times:
Here a is called the base and n the exponent. We say that a is raised to the nth power.
Note that a1=a, since we have 1×a=a.
Any number raised to the power zero is
And since any number multiplied by 0 is 0:
When n=0, we have 00, 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:
If two exponentials in the same positive base are equal, their exponents must be equal:
Exponentials can also appear in equations with one or more unknown:
Now we can equate the exponents:
Nice work completing Exponentiation and integer exponent laws, here's a quick recap of what we covered:
Exercises checked off