Lesson 4.14
Solving with Rational Exponents
Some equations use rational (fractional) exponents instead of radical notation. The solving strategy: raise both sides to the reciprocal power to cancel the exponent.
Introduction
An equation like is solved by raising both sides to the power (the reciprocal of ). This method is cleaner than converting to radical form and follows logically from exponent rules.
Past Knowledge
Rational exponents (Ch. 14), solving radical equations (4.13), exponent rules.
Today's Goal
Solve equations involving rational exponents using the reciprocal power method.
Future Success
4.15 covers extraneous solutions that can arise with even-denominator exponents.
Key Concepts
The Reciprocal Power Method
If , raise both sides to :
Even vs. Odd Denominator
Odd denominator
One solution:
Even denominator
Consider ± : (but check domain!)
Worked Examples
Example 1: Unit Fraction Exponent
BasicSolve .
Raise both sides to the reciprocal power (3)
Check
Example 2: General Rational Exponent
IntermediateSolve .
Raise to the power
But also consider the negative root
So
Example 3: Isolate First
AdvancedSolve .
Isolate the power expression
Raise to
Solve
Common Pitfalls
Forgetting the ± Case
When the numerator of the exponent is even, both positive and negative values can work. Always check both possibilities.
Wrong Reciprocal
The reciprocal of is , not . Flip the fraction.
Real-Life Applications
In physiology, the relationship between body mass and metabolic rate follows (Kleiber's Law). Solving for mass given a metabolic rate requires the reciprocal power method.
Practice Quiz
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