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

검증된 단계별 안내

Start with the given equation: \( (x-4)^{\frac{2}{5}} = 9 \).

To eliminate the fractional exponent, raise both sides of the equation to the reciprocal power, which is \( \frac{5}{2} \), so you get: \( \left((x-4)^{\frac{2}{5}}\right)^{\frac{5}{2}} = 9^{\frac{5}{2}} \).

Simplify the left side using the property of exponents \( (a^{m})^{n} = a^{mn} \), which gives: \( (x-4)^{\frac{2}{5} \times \frac{5}{2}} = (x-4)^1 = x-4 \).

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.

Finally, solve for \( x \) by adding 4 to both sides: \( x = 9^{\frac{5}{2}} + 4 \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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