Solve each equation. (x-4)^2/5 = 9

Verified step by step guidance

1. Start with the given equation: \(\frac{(x-4)^2}{5} = 9\).
2. Multiply both sides of the equation by 5 to eliminate the denominator: \((x-4)^2 = 9 \times 5\).
3. Simplify the right side: \((x-4)^2 = 45\).
4. Take the square root of both sides to solve for \(x-4\): \(x-4 = \pm \sqrt{45}\).
5. Simplify the square root if possible, then solve for \(x\) by adding 4 to both sides: \(x = 4 \pm \sqrt{45}\).

Key Concepts

Solving Rational Exponent Equations

Equations involving rational exponents can be solved by isolating the term with the exponent and then raising both sides of the equation to the reciprocal power. For example, if an expression is raised to the 2/5 power, raising both sides to the 5/2 power will eliminate the exponent.

Isolating the Variable

Before applying exponent rules, it is important to isolate the expression containing the variable. This means rewriting the equation so that the term with the variable is alone on one side, making it easier to apply inverse operations and solve for the variable.

Checking for Extraneous Solutions

When solving equations involving even roots or rational exponents, some solutions may not satisfy the original equation. It is essential to substitute solutions back into the original equation to verify their validity and discard any extraneous solutions.