Domanda del libro di testo
In Exercises 48–57, perform the indicated operations and write the result in standard form. 6/(5+i)
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 2, Problema 52
Solve each equation in Exercises 41–60 by making an appropriate substitution.
Guida verificata passo dopo passo1
Identify the substitution to simplify the equation. Notice that the exponents are fractional and related: \(x^{2/5}\) and \(x^{1/5}\). Let \(u = x^{1/5}\), so that \(u^2 = x^{2/5}\).
Rewrite the original equation in terms of \(u\): replace \(x^{2/5}\) with \(u^2\) and \(x^{1/5}\) with \(u\). The equation becomes \(u^2 + u - 6 = 0\).
Solve the quadratic equation \(u^2 + u - 6 = 0\) using factoring, completing the square, or the quadratic formula. This will give you the possible values of \(u\).
After finding the values of \(u\), recall that \(u = x^{1/5}\). To find \(x\), raise both sides of the equation \(u = x^{1/5}\) to the 5th power, resulting in \(x = u^5\).
Calculate \(x\) for each value of \(u\) found in step 3 by computing \(x = u^5\). These are the solutions to the original equation.
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Exponents and Rational Powers
Rational exponents represent roots and powers, such as x^(m/n) meaning the n-th root of x raised to the m-th power. Understanding how to manipulate and simplify expressions with fractional exponents is essential for rewriting and solving equations involving these terms.
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04:06
Rational Exponents
Substitution Method
Substitution involves replacing a complex expression with a simpler variable to transform the equation into a more familiar form, often a polynomial. This technique simplifies solving equations that contain complicated terms, such as fractional powers, by reducing them to quadratic or linear equations.
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04:03Choosing a Method to Solve Quadratics
Solving Quadratic Equations
Once substitution reduces the equation to a quadratic form, solving it involves finding values of the variable that satisfy the equation using factoring, completing the square, or the quadratic formula. Understanding how to solve quadratics is crucial to find the solutions to the original equation after back-substitution.
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06:08Solving Quadratic Equations by Factoring
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