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

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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