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Rational and Irrational Numbers and Their PropertiesAdapted from Walch Education

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.4.1.2: Rational and Irrational Numbers and Their Properties2Key Concepts

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. 4.1.2: Rational and Irrational Numbers and Their Properties3Key concepts, continued In the following equations, x is the variable and a, b, m, and n represent integers, with m 0 and n 0.4.1.2: Rational and Irrational Numbers and Their Properties4Key concepts, continuedPower:Root:

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. 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.

4.1.2: Rational and Irrational Numbers and Their Properties5ImportantSimplify the expression 4.1.2: Rational and Irrational Numbers and Their Properties6Practice

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.4.1.2: Rational and Irrational Numbers and Their Properties7Apply the Product of Powers Property to simplify the expression

Simplify the expression 4.1.2: Rational and Irrational Numbers and Their Properties8You try!

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