Understanding Rational Exponents in Mathematics
Rational exponents, represented by fractions, follow similar properties as integer exponents. This chapter delves into definitions, radical notation, rewriting expressions with rational exponents, and simplifying complex expressions. Through practical examples and explanations, readers can grasp the concept of rational exponents effectively.
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Chapter 7 Section 2 Rational Exponents Page 515
What are Rational Exponents? Exponents that are fractions The properties of rational exponents are the same for properties for integer exponents.
1 n a Definitions of n If represents a real number and n 2 is an integer then a 1 n= a n a If n is even, a must be nonnegative. If n is odd, a can be any real number. Note: note the denominator and the index.
Radical Notation, Exponential Notation Radical Notation: Radical sign is used. Example: 3?? Exponential Notation: Exponents are used. 1 3 Example: 4
Using the definition Use radical notation to rewrite the expression. Simplify, if possible 1 2 64 1 5 ( ) 6x2y
Check the definition again. 1 n= a 1 n =a n n a a or Radical s index becomes the exponent s denominator. Radicand become the base.
Rewrite with rational exponents 5 13ab a) xy2 17 7 b)
Observe 2 ( ) 3( ) 3 1 3 2 3= a 1 3 what do you notice? a 2 3= a2 a 2 3= 2 3= a2 2 a a a
m n Definition of a n( ) m n= m a a and m n= am n a Numerator of the exponent become the exponent of the base Denominator of the exponent become the root index.
Use the definition Use radical notation to rewrite and simplify. 2 3 1000 3 5 -32
Rewrite with rational exponents 3 75 4( ) 9 13xy
Rewrite with positive exponents and simplify -1 36 2 -4 ( ) 7xy 7
An expression with rational exponents is simplified when No parentheses appear. No powers are raised to powers. Each base occurs only once. No negative or zero exponents appear.
Simplify completely 1 7 6 4 7 6 a) 1 2 -2 1 3 x 5y b) 1 2 32x c) 16x 3 4
Simplify completely. Use rational exponents 10 x5 a) x6y2 3 b) 3 x x c)
Summary Rational exponent exponent denominator is the root index ? ?= radical expression root index is the denominator of the fractional exponent ? ? ??? ? ???= ?
