Introduction The properties of integer exponents also apply to
irrational exponents. In this section, we will see how the
following properties can be used to write equivalent exponential
expressions involving irrational exponents. We will also learn how
to tell if a sum or product will be rational or irrational, and we
will solve equations that have rational exponents as well as
equations that have irrational exponents. 1 4.1.2: Rational and
Irrational Numbers and Their Properties
Slide 2
Key Concepts An irrational number is a number that cannot be
expressed as the ratio of two integers. In other words, it cannot
be written as a fraction that has whole numbers for both the
numerator and denominator. The decimal representations of
irrational numbers neither end nor repeat. The root of a number for
which there is no exact root is an irrational number, such as
Addition and multiplication are closed operations within integers,
meaning that the sum of any integers is an integer, and the product
of any integers is also an integer. 2 4.1.2: Rational and
Irrational Numbers and Their Properties
Slide 3
Key Concepts, continued Equations involving a power can be
solved by taking a root, and equations involving a root can be
solved by raising to a power. In order to solve an equation this
way, one side of the equation must be a single variable and the
other side must be a quantity. If the exponent is rational, raise
both sides to the multiplicative inverse (reciprocal) of the
fraction. 3 4.1.2: Rational and Irrational Numbers and Their
Properties
Slide 4
Key Concepts, continued If the expression is written using a
power and a root, it can be rewritten using a rational exponent and
then solved using the multiplicative inverse of the rational
fraction. Or, each operation, the root and the power, can be undone
in separate steps. The same operation must be performed on both
sides of the equation. 4 4.1.2: Rational and Irrational Numbers and
Their Properties
Slide 5
Key Concepts, continued In the following equations, x is the
variable and a, b, m, and n represent integers, with m 0 and n 0. 5
4.1.2: Rational and Irrational Numbers and Their Properties
Power:Root:
Slide 6
Key Concepts, continued 6 4.1.2: Rational and Irrational
Numbers and Their Properties Rational exponent case 1: Rational
exponent case 2:
Slide 7
Key Concepts, continued The sum of two rational numbers is a
rational number. The product of two rational numbers is a rational
number. The sum of a rational number and an irrational number is an
irrational number. The product of a rational number and an
irrational number is an irrational number. 7 4.1.2: Rational and
Irrational Numbers and Their Properties
Slide 8
Common Errors/Misconceptions incorrectly categorizing a root as
irrational when an exact root exists incorrectly categorizing a
repeating decimal as irrational incorrectly raising a power to a
power when solving an exponential equation trying to solve an
exponential equation without first isolating the variable confusing
the rules for the Power of a Power Property, the Power of a Product
Property, and the Product of Powers Property 8 4.1.2: Rational and
Irrational Numbers and Their Properties
Slide 9
Guided Practice Example 1 Simplify the expression 9 4.1.2:
Rational and Irrational Numbers and Their Properties
Slide 10
Guided Practice: Example 1, continued 1.Identify which property
can be used to simplify the expression. This is the product of two
exponential expressions with the same base. Use the Product of
Powers Property to simplify. 10 4.1.2: Rational and Irrational
Numbers and Their Properties
Slide 11
Guided Practice: Example 1, continued 2.Apply the property to
simplify the expression. The Product of Powers Property states that
if the bases are the same, the expression can be written as the
single base raised to the sum of the powers. 11 4.1.2: Rational and
Irrational Numbers and Their Properties
Slide 12
Guided Practice: Example 1, continued 12 4.1.2: Rational and
Irrational Numbers and Their Properties
Slide 13
Guided Practice Example 2 Simplify the expression 13 4.1.2:
Rational and Irrational Numbers and Their Properties
Slide 14
Guided Practice: Example 2, continued 1.Identify which property
can be used to simplify the expression. This is the quotient of two
exponential expressions with the same base. Use the Quotient of
Powers Property to simplify. 14 4.1.2: Rational and Irrational
Numbers and Their Properties
Slide 15
Guided Practice: Example 2, continued 2.Apply the property to
simplify the expression. The Quotient of Powers Property states
that since the bases are the same, the expression can be written as
the base raised to the difference between the power of the
numerator and the power of the denominator. If the exponent is
negative, the Negative Exponent Property can be used to rewrite the
expression. 15 4.1.2: Rational and Irrational Numbers and Their
Properties
Slide 16
Guided Practice: Example 2, continued 16 4.1.2: Rational and
Irrational Numbers and Their Properties