Question

Solve each equation. $$ \sqrt[3]{2 m-1}=\sqrt[3]{m+13} $$

   Solve each equation.
$$
\sqrt[3]{2 m-1}=\sqrt[3]{m+13}
$$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Margaret L. Lial ,… 5th Edition

Chapter 10, Problem 41

Instant Answer

Solved by Expert James Kiss

07/13/2020

Step 1

This gives us: $$ (2m-1)^3 = (m+13)^3 $$ which simplifies to: $$ 2m - 1 = m + 13 $$ Show more…

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Solve each equation. $$ \sqrt[3]{2 m-1}=\sqrt[3]{m+13} $$

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Key Concepts

Cube Roots

Cube roots are the inverses of cubing functions that allow for extracting a number which, when raised to the third power, gives the original value. They are defined for all real numbers and are essential in solving equations involving radicals.

One-to-One Functions

A one-to-one function is a function where each output is produced by exactly one input. The cube root function is one-to-one, which means if two cube roots are equal, then their radicands (the expressions under the radical) must also be equal, enabling simplification of the equation.

Inverse Operations

Inverse operations are used to 'undo' a mathematical operation. In the context of cube roots, cubing both sides of an equation acts as the inverse operation to taking the cube root, thus eliminating the radical and simplifying the problem.

Solving Linear Equations

Once the radicals are removed, the equation typically simplifies to a linear form. Solving linear equations involves combining like terms and systematically isolating the variable using arithmetic operations, which is a foundational technique in algebra.

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