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

Verified step by step guidance

1. Start with the given equation: \( (x-3)^{\frac{2}{5}} = 4 \). Our goal is to solve for \(x\).
2. To eliminate the fractional exponent, raise both sides of the equation to the reciprocal power of \(\frac{2}{5}\), which is \(\frac{5}{2}\). This gives: \(\left((x-3)^{\frac{2}{5}}\right)^{\frac{5}{2}} = 4^{\frac{5}{2}}\).
3. Simplify the left side using the property of exponents \(\left(a^{m}\right)^{n} = a^{mn}\), so \( (x-3)^{\frac{2}{5} \times \frac{5}{2}} = (x-3)^1 = x-3 \).
4. Now the equation is \( x - 3 = 4^{\frac{5}{2}} \). Next, express \(4^{\frac{5}{2}}\) as \(\left(4^{\frac{1}{2}}\right)^5\) or \(\left(\sqrt{4}\right)^5\) to simplify the right side.
5. Finally, solve for \(x\) by adding 3 to both sides: \( x = 3 + 4^{\frac{5}{2}} \). This gives the solution for \(x\).

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 these exponents is essential for solving equations involving fractional powers.

Isolating the Variable Expression

Before solving, isolate the expression with the variable on one side of the equation. This step simplifies the equation and prepares it for applying inverse operations, such as raising both sides to a reciprocal power to eliminate the rational exponent.

Checking for Extraneous Solutions

When solving equations with rational exponents, especially involving even roots, some solutions may not satisfy the original equation. Substituting solutions back into the original equation ensures only valid answers are accepted.