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

Verified step by step guidance

1. Start with the given equation: \(\frac{(x-3)^2}{5} = 4\).
2. Multiply both sides of the equation by 5 to eliminate the denominator: \((x-3)^2 = 4 \times 5\).
3. Simplify the right side: \((x-3)^2 = 20\).
4. Take the square root of both sides to solve for \(x-3\): \(x-3 = \pm \sqrt{20}\).
5. Solve for \(x\) by adding 3 to both sides: \(x = 3 \pm \sqrt{20}\).

Key Concepts

Solving Rational Exponent Equations

Equations involving variables raised to rational exponents can be solved by isolating the term with the exponent and then raising both sides of the equation to the reciprocal power. This process eliminates the rational exponent and simplifies the equation to a polynomial or linear form.

Properties of Exponents

Understanding how to manipulate exponents is essential, including the rule that (a^m)^n = a^(m*n). This allows you to raise both sides of an equation to a power to eliminate fractional exponents and solve for the variable inside the base.

Checking for Extraneous Solutions

When solving equations involving rational exponents or radicals, it is important to check all solutions in the original equation. Raising both sides to a power can introduce extraneous solutions that do not satisfy the original equation.