Solve each equation. See Example 7.(x-3)^2/5=(4x)^1/5

Verified step by step guidance

1. Start by recognizing that both sides of the equation have expressions raised to the power of \( \frac{1}{5} \). The equation is \( (x-3)^{\frac{2}{5}} = (4x)^{\frac{1}{5}} \).
2. To simplify, raise both sides of the equation to the 5th power to eliminate the fractional exponents. This gives \( \left((x-3)^{\frac{2}{5}}\right)^5 = \left((4x)^{\frac{1}{5}}\right)^5 \).
3. Using the power of a power property \( (a^{m})^{n} = a^{mn} \), simplify both sides: \( (x-3)^2 = 4x \).
4. Rewrite the equation as \( (x-3)^2 = 4x \) and expand the left side using the binomial formula: \( (x-3)^2 = x^2 - 6x + 9 \).
5. Set up the quadratic equation by moving all terms to one side: \( x^2 - 6x + 9 = 4x \) becomes \( x^2 - 10x + 9 = 0 \). Then, solve this quadratic equation using factoring, completing the square, or the quadratic formula.

Key Concepts

Properties of Exponents and Radicals

Understanding how to manipulate expressions with fractional exponents is essential. A fractional exponent like 1/5 represents the fifth root, so (x-3)^(2/5) means the fifth root of (x-3) squared. Recognizing this helps in rewriting and simplifying the equation.

Solving Equations Involving Radicals

To solve equations with radicals or fractional exponents, one often isolates the radical expression and then raises both sides to a power that eliminates the root. This process requires careful handling to avoid extraneous solutions and to maintain equivalence.

Checking for Extraneous Solutions

When both sides of an equation are raised to powers to eliminate radicals, new solutions may appear that do not satisfy the original equation. It is important to substitute solutions back into the original equation to verify their validity.