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

Verified step by step guidance

1. Start with the given equation: \( (x-4)^{\frac{2}{5}} = 9 \).
2. To eliminate the fractional exponent, raise both sides of the equation to the reciprocal power, which is \( \frac{5}{2} \), so you get: \( \left((x-4)^{\frac{2}{5}}\right)^{\frac{5}{2}} = 9^{\frac{5}{2}} \).
3. Simplify the left side using the property of exponents \( (a^{m})^{n} = a^{mn} \), which gives: \( (x-4)^{\frac{2}{5} \times \frac{5}{2}} = (x-4)^1 = x-4 \).
4. Now the equation is \( x - 4 = 9^{\frac{5}{2}} \). Calculate \( 9^{\frac{5}{2}} \) by first finding the square root of 9, then raising the result to the 5th power.
5. Finally, solve for \( x \) by adding 4 to both sides: \( x = 9^{\frac{5}{2}} + 4 \).

Key Concepts

Rational Exponents

Rational exponents represent roots and powers simultaneously. For example, an exponent of 2/5 means raising the base to the power 2 and then taking the fifth root, or vice versa. Understanding how to manipulate and interpret these exponents is essential for solving equations involving fractional powers.

Isolating the Variable Expression

Before solving, isolate the expression containing the variable, such as (x - 4)^(2/5). This allows you to apply inverse operations, like raising both sides to the reciprocal power, to eliminate the rational exponent and simplify the equation for easier solving.

Solving Equations with Even Powers

When an expression is raised to an even power, the equation may have two solutions: one positive and one negative. After isolating the base, consider both the positive and negative roots when solving for the variable to ensure all possible solutions are found.