Domanda del libro di testo
In Exercises 39–48, factor the difference of two squares. 16x4−81
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Ch. P - Fundamental Concepts of Algebra
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 1, Problema 47
Rationalize the denominator.
Guida verificata passo dopo passo1
Identify the expression to rationalize: \(\frac{\sqrt{2}}{\sqrt{5}}\). The goal is to eliminate the square root from the denominator.
Multiply both the numerator and the denominator by \(\sqrt{5}\), which is the conjugate in this case, to rationalize the denominator. This gives: \(\frac{\sqrt{2}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\).
Use the property of square roots that \(\sqrt{a} \times \sqrt{a} = a\) to simplify the denominator: \(\sqrt{5} \times \sqrt{5} = 5\).
Multiply the numerators together: \(\sqrt{2} \times \sqrt{5} = \sqrt{2 \times 5} = \sqrt{10}\).
Write the new expression with the rationalized denominator: \(\frac{\sqrt{10}}{5}\). This is the simplified form with no radical in the denominator.
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Rationalizing the Denominator
Rationalizing the denominator involves eliminating any radicals (square roots) from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable expression that will remove the radical from the denominator, making the expression easier to interpret and use.
Video consigliato:
02:58
Rationalizing Denominators
Properties of Square Roots
Square roots have properties such as \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \) and \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). Understanding these properties helps simplify expressions involving radicals and is essential when rationalizing denominators.
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02:20Imaginary Roots with the Square Root Property
Multiplying by a Form of One
To rationalize a denominator, multiply the fraction by a form of one that contains the radical in the denominator, such as \( \frac{\sqrt{5}}{\sqrt{5}} \). This does not change the value of the expression but helps eliminate the radical from the denominator.
Video consigliato:
04:15Multiply Polynomials Using the Distributive Property
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