Question

Solve by factoring. $$t^{5 / 3}-25 t^{-1 / 3}=0$$

   Solve by factoring.
$$t^{5 / 3}-25 t^{-1 / 3}=0$$

Precalculus

Cynthia Young 1st Edition

Chapter 0, Problem 79

Instant Answer

Solved by Expert Christy Galilei

06/16/2021

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This gives us: $$t^{-1/3}(t^{6/3}-25)=0$$ Show more…

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Solve by factoring. $$t^{5 / 3}-25 t^{-1 / 3}=0$$

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Key Concepts

Factoring Techniques

Factoring is the process of rewriting an algebraic expression as a product of simpler expressions. This method is crucial when solving equations because it allows one to apply further properties, such as the zero product property, to identify the solutions of the equation.

Properties of Exponents

Understanding and applying the laws of exponents, especially for fractional and negative exponents, is essential for simplifying expressions. Fractional exponents represent roots and negative exponents indicate reciprocals, enabling one to manipulate and combine terms effectively during the factoring process.

Zero Product Property

The zero product property states that if a product of factors equals zero, then at least one of the factors must be zero. This principle is fundamental in solving factored equations, as it reduces the original equation into simpler equations that can be solved individually.

Domain Considerations in Algebraic Equations

When dealing with expressions that have negative or fractional exponents, it is important to consider the permissible values of the variable (the domain). This ensures that operations such as taking reciprocals and roots do not lead to undefined expressions, thus helping to avoid extraneous solutions.

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