Sections 5.1 and 5.2
Exponents
﴾We are not doing Scientific Notation﴿
Objectives:
• Use the product rule for exponents
• Evaluate expressions raised to the zero power
• Use the quotient rule for exponents
• Evaluate expressions raised to the negative nth power
• Use the power rule for exponents
• Use exponent rules to and definitions to simplify
exponential expressions
Exponents:
Exponents: a shorthand way of writing...
x is the
and 4 is the
It means:
Exponents:
Product rule for exponents:
If m and n are positive integers and a is a real number, then:
Or:
To multiply two powers with the same base, add their exponents.
The Product Rule for Exponents:
Product rule for exponents:
If m and n are positive integers and a is a real number, then:
Only simplify one thing at a time—and show every step. You’ll make fewer mistakes.
Exponents:
Power Rules for Exponents:
If a and b are real numbers and m and n are integers, then:
Power rule
Power of a product
Power of a quotient
Or:
To raise a power to a power, multiply the exponents.
To raise a product/quotient to a power, distribute the exponents.
The Power Rule for Exponents:
Power Rule for Exponents:
If a and b are real numbers and m and n are integers, then:
Remember:
Only simplify one thing at a time—and show every step. You’ll make fewer mistakes.
Exponents:
Quotient Rule for Exponents:
If a is a nonzero real number and m and n are both integers, then:
Or:
To divide powers with the same base, subtract the exponent in the denominator
from the exponent in the numerator.
The Quotient Rule for Exponents:
Quotient Rule for Exponents:
If a is a nonzero real number and
m and n are both integers, then:
Exponents:
Zero Exponent:
If a does not equal zero then:
Zero Exponent:
Zero Exponent:
If a does not equal zero then:
Exponents:
Negative Exponents:
If a is a real number other than 0 and n is a positive integer, then:
Or:
To raise a base to a negative power, write the reciprocal of the base and
make the exponent positive.
Negative Exponents:
Negative Exponents:
If a is a real number other than 0 and n is a positive integer, then:
Only negative exponents move, not negative coefficients
Negative Exponents:
Negative Exponents:
If a is a real number other than 0 and n is a positive integer, then:
Only negative exponents move, not negative coefficients
The Properties of Exponents:
If a and b are real numbers and m and n are integers, then:
Product rule
Zero Exponent
Negative Exponent
Quotient Rule
Power rule
Power of a product
Power of a quotient
Practice Exponents:
Simplify completely. Write with positive exponents only.
Simplifying Exponents:
The Rule
No more negative exponents
The numbers in the fraction
are simplified
Numbers raised to powers
are evaluated
Zero should not be seen as
an exponent
If variables appear more than
once, combine them
No more parentheses
Factors should be combined
Not Yet Simplified
Simplified