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Solve: √(6x - 2) = √(2x + 3) - √(4x - 1).
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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 143
Solve for x:
Guida verificata passo dopo passo1
Rewrite the expression inside the cube root using exponents. Recall that \( \sqrt{x} = x^{\frac{1}{2}} \), so \( x \sqrt{x} = x \cdot x^{\frac{1}{2}} = x^{1 + \frac{1}{2}} = x^{\frac{3}{2}} \).
Express the cube root as an exponent: \( \sqrt[3]{x^{\frac{3}{2}}} = \left(x^{\frac{3}{2}}\right)^{\frac{1}{3}} \).
Use the power of a power property to simplify the exponent: \( \left(x^{\frac{3}{2}}\right)^{\frac{1}{3}} = x^{\frac{3}{2} \cdot \frac{1}{3}} = x^{\frac{1}{2}} \).
Set the simplified expression equal to 9: \( x^{\frac{1}{2}} = 9 \).
To solve for \( x \), square both sides to eliminate the fractional exponent: \( \left(x^{\frac{1}{2}}\right)^2 = 9^2 \), which simplifies to \( x = 81 \).
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Radical Expressions and Their Properties
Radical expressions involve roots such as square roots and cube roots. Understanding how to manipulate and simplify these expressions, including converting between radicals and exponents, is essential for solving equations involving roots.
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Radical Expressions with Fractions
Exponents and Fractional Powers
Exponents can be fractions, where the numerator represents the power and the denominator the root. For example, x^(m/n) means the nth root of x raised to the mth power. This concept helps rewrite radicals as exponents to simplify and solve equations.
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Solving Equations Involving Radicals
To solve equations with radicals, isolate the radical expression and then eliminate the root by raising both sides to the appropriate power. This process often requires careful handling of exponents and checking for extraneous solutions.
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5:02Solving Logarithmic Equations
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