Question

Solve. $$ \sqrt[3]{4 c^{2}-5 c+11}=\sqrt[3]{c^{2}+9} $$

   Solve.
$$
\sqrt[3]{4 c^{2}-5 c+11}=\sqrt[3]{c^{2}+9}
$$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Sherri Messersmith 3rd Edition

Chapter 10, Problem 76

Instant Answer

Solved by Expert Narayan Hari

02/19/2021

Step 1

To simplify this, we can cube both sides of the equation to eliminate the cube root. This gives us: $$ 4 c^{2}-5 c+11 = c^{2}+9 $$ Show more…

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Solve. $$ \sqrt[3]{4 c^{2}-5 c+11}=\sqrt[3]{c^{2}+9} $$

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Key Concepts

Cube Root Properties

Cube roots are the inverse operation of cubing and unlike square roots, cube roots are defined for all real numbers. Understanding their properties is essential because applying the cube operation can simplify equations by removing the radical, leading to a straightforward polynomial equation.

Inverse Operations - Cubing Both Sides

Applying the cube operation to both sides of an equation that contains cube roots is an inverse operation that eliminates the radical signs. This step transforms the original equation into a polynomial equation that is easier to solve, but it is important to note that this operation preserves equality because cube roots have a one-to-one correspondence with their cube values.

Solving Polynomial Equations

Once the cube roots are eliminated, the equation often reduces to a polynomial equation. Solving such equations requires standard algebraic techniques like combining like terms, factoring, or using the quadratic formula. Understanding these techniques is crucial for finding the correct solutions to the original equation.

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