Textbook Question
Write each statement using an absolute value equation or inequality.
m is no more than 2 units from 7.
Ch. 1 - Equations and Inequalities
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 2, Problem 69
Solve each equation. (x-2)2/3 = x1/3
Verified step by step guidance1
Rewrite the equation \((x-2)^{2/3} = x^{1/3}\) to make it easier to work with. Notice that both sides have fractional exponents with denominator 3, so consider raising both sides to the power of 3 to eliminate the cube roots.
Raise both sides of the equation to the power of 3 to clear the fractional exponents: \(\left((x-2)^{2/3}\right)^3 = \left(x^{1/3}\right)^3\). Simplify the exponents by multiplying: \((x-2)^2 = x\).
Expand the left side of the equation: \((x-2)^2 = x^2 - 4x + 4\). So the equation becomes \(x^2 - 4x + 4 = x\).
Bring all terms to one side to set the equation equal to zero: \(x^2 - 4x + 4 - x = 0\), which simplifies to \(x^2 - 5x + 4 = 0\).
Solve the quadratic equation \(x^2 - 5x + 4 = 0\) by factoring, completing the square, or using the quadratic formula to find the possible values of \(x\). Remember to check each solution in the original equation because of the fractional exponents.
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6mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Exponents
Rational exponents represent roots and powers simultaneously. For example, x^(m/n) means the n-th root of x raised to the m-th power. Understanding how to manipulate and simplify expressions with rational exponents is essential for solving equations like (x-2)^(2/3) = x^(1/3).
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Rational Exponents
Isolating and Equating Expressions
To solve equations involving exponents, it is often necessary to isolate terms and rewrite the equation so that bases or exponents can be compared or equated. This may involve raising both sides to a power to eliminate fractional exponents or rewriting expressions with a common base.
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Checking for Extraneous Solutions
When solving equations with rational exponents, especially involving even roots, some solutions may not satisfy the original equation. It is important to substitute solutions back into the original equation to verify their validity and exclude any extraneous solutions.
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Related Practice
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