Solve each equation. See Examples 8 and 9. 2x^-2/5-x^-1/5-1=0

검증된 단계별 안내

Rewrite the equation \(2x^{-\frac{2}{5}} - x^{-\frac{1}{5}} - 1 = 0\) by recognizing the negative exponents as reciprocals. For example, \(x^{-\frac{2}{5}} = \frac{1}{x^{\frac{2}{5}}}\) and \(x^{-\frac{1}{5}} = \frac{1}{x^{\frac{1}{5}}}\).

To simplify the equation, introduce a substitution: let \(y = x^{-\frac{1}{5}}\). Then, \(x^{-\frac{2}{5}} = (x^{-\frac{1}{5}})^2 = y^2\). Rewrite the equation in terms of \(y\).

Substitute into the equation to get a quadratic form: \(2y^2 - y - 1 = 0\). This is a standard quadratic equation in \(y\).

Solve the quadratic equation \(2y^2 - y - 1 = 0\) using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=2\), \(b=-1\), and \(c=-1\).

After finding the values of \(y\), substitute back \(y = x^{-\frac{1}{5}}\) and solve for \(x\) by raising both sides to the power of \(-5\) to isolate \(x\).

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주요 개념

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

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, x^(-n) equals 1 divided by x^n. Understanding this helps rewrite terms like x^(-2/5) as 1 over x^(2/5), simplifying the equation.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously; x^(m/n) means the n-th root of x raised to the m-th power. For instance, x^(1/5) is the fifth root of x. Recognizing this allows manipulation of terms with fractional powers effectively.

Solving Rational Equations

Solving equations involving variables with negative and fractional exponents often requires rewriting terms to a common base or exponent, then isolating the variable. Techniques include substitution or multiplying through by an expression to clear denominators, enabling solution of the equation.

Solve each equation. See Examples 8 and 9. 2x-2/5-x-1/5-1=0