Lesson 3 MA 152
Definition of a Rational Exponent (1):
If then is the real number b such that , if b exists. Basically,.
Definition of a root: If n is a natural number greater than 1 and , then is the nonnegative number whose nth power is a.
If n is a natural number greater than 1 and .
Ex 1: Evaluate each.
Sometimes an absolute value symbol must be used, if the variable is not defined as a positive value whenever there is an even power. However, we will assume that each variable represents a positive.
Ex 2: Evaluate each.
Other Rational Exponents (2):
Definition: If m and n are positive integers, the fraction is in lowest terms, and exists, then . This definition shows that can be evaluated two different ways; take the nth root first, then raise to the mth power or raise to the mth power first, then take the nth root. It is usually to the student's advantage to take the root first, then raise to the power.
Ex 1: Evaluate each.
All the rules of exponents previously studied now apply also with rational exponents.
Ex 2: Assume that all variables represent positive integers. Evaluate or simplify each and write all answers without using negative exponents.
Ex 3: Simplify each using the rules of exponents. Assume that all variables represent positive integers. Write answers without using negative exponents.
In the radical expression , the symbol is called the radical sign, a is the radicand, and n is the index. (If the index is 2, a square root, usually the index is not written. If the index is 3, the expression is called a cube root. Otherwise, it is an nth root.)
If n is a natural number greater than 1 and , then is the nonnegative number whose nth power is a.
If .
If a < 0, and
Ex 4: Evaluate each.
Multiplication and Division Properties of Radicals:
If all expressions represent real numbers,
Note: These properties are for multiplication and division. Similar statements are not true for addition or subtraction. ( for example)
Numbers that are squares of positive integers; such as 1, 4, 9, 16, ...; are called perfect squares. Powers such as , ... are also called perfect squares.
Numbers that are cubes of positive integers; such as 1, 8, 27, 64,....; are called perfect cubes. Powers such as are also called perfect cubes.
To simplify a radical expression:
- Factor the radicand into the greatest perfect factor possible and a 'leftover' factor.
- Use the multiplication property of radicals to 'separate' into two radicals.
- Simplify the 'perfect factor' radical.
Ex 5: Simplify each.
Racial expressions with the same index and the same radicand are called like radicals. Like radicals can be combined similarly as like terms. *When trying to combine radicals, always simplify radicals first!
Ex 6: Simplify and combine where possible.
Rationalizing Denominators: Rationalizing a denominator means to write the expression without a radical in the denominator. To rationalize when there is a square root in the denominator do the following steps.
- First, simplify each radical.
- Multiply numerator and denominator by the radical factor in the denominator.
- Simplify numerator and denominator.
Ex 7: Rationalize the denominator.