Question

Solve the equation. $$ 3 y^{2 / 3}+2 y^{1 / 3}-8=0 $$

   Solve the equation. 
$$
3 y^{2 / 3}+2 y^{1 / 3}-8=0
$$

Precalculus

Raymond A. Barnett,… 7th Edition

Chapter 1, Problem 40

Instant Answer

Solved by Expert Brandon Cleary

07/16/2021

Step 1

We can make the substitution $u = y^{1/3}$. Substituting this in, we get: $$ 3u^2 + 2u - 8 = 0 $$ Show more…

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Solve the equation. $$ 3 y^{2 / 3}+2 y^{1 / 3}-8=0 $$

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Key Concepts

Quadratic Equation

Many algebraic problems reduce to a quadratic equation after an appropriate substitution. A quadratic equation, generally expressed in the form ax² + bx + c = 0, can be solved using techniques like factoring, completing the square, or applying the quadratic formula. Recognizing and solving quadratic equations is a fundamental skill in algebra.

Fractional Exponents

Fractional exponents allow expressions like roots to be written in exponential form. For instance, writing the cube root of a variable as raising it to the 1/3 power facilitates the use of exponent rules and simplifies manipulation in algebraic contexts.

Substitution Method

The substitution method involves replacing a complicated expression with a simpler variable, making the equation easier to solve. When an equation features repeated expressions, such as a variable raised to different fractional powers, substituting one of these expressions can transform the equation into a more recognizable form.

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