Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Linear Equations

College Algebra

1. Equations & Inequalities

Linear Equations

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1:37 minutes

Problem 76

Textbook Question

Solve each equation. ⁵√ 2x = ⁵√ 3x+2

Verified step by step guidance

Recognize that both sides of the equation involve fifth roots, so you can eliminate the roots by raising both sides to the power of 5.

Rewrite the equation as $(\sqrt[5]{2x})^5 = (\sqrt[5]{3x+2})^5$.

Simplify both sides to get $2x = 3x + 2$.

Rearrange the equation to isolate $x$ by subtracting $3x$ from both sides, resulting in $2x - 3x = 2$.

Simplify the equation to $-x = 2$, and then multiply both sides by $-1$ to solve for $x$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radical Equations

Radical equations involve variables within a radical symbol, such as square roots or cube roots. To solve these equations, one typically isolates the radical on one side and then raises both sides of the equation to the power that eliminates the radical. Understanding how to manipulate these equations is crucial for finding the values of the variable.

Properties of Exponents

Properties of exponents are rules that govern how to handle expressions involving powers. For instance, when dealing with roots, the nth root of a number can be expressed as a fractional exponent (e.g., ⁵√x = x^(1/5)). This concept is essential for rewriting radical expressions in a more manageable form, facilitating easier calculations.

Isolating Variables

Isolating variables is a fundamental algebraic technique used to solve equations. It involves rearranging the equation to get the variable of interest on one side by performing inverse operations. This concept is vital in solving equations, especially when dealing with multiple terms or radicals, as it allows for a clearer path to finding the solution.