Textbook Question
In Exercises 65–92, factor completely, or state that the polynomial is prime. x3+2x2−9x−18
Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 1, Problem 72
Simplify by reducing the index of the radical :
Verified step by step guidance1
Recognize that the expression \( [y^3]^{1/6} \) is a radical expression written with rational exponents. The exponent \( \frac{1}{6} \) represents the sixth root.
Use the property of exponents that \( (a^m)^n = a^{m \times n} \) to combine the powers inside the expression.
Multiply the exponents: \( 3 \times \frac{1}{6} = \frac{3}{6} \). So, the expression becomes \( y^{\frac{3}{6}} \).
Simplify the fraction \( \frac{3}{6} \) to its lowest terms, which is \( \frac{1}{2} \). Now the expression is \( y^{\frac{1}{2}} \).
Recognize that \( y^{\frac{1}{2}} \) is equivalent to the square root of \( y \), or \( \sqrt{y} \). This is the simplified form with the reduced index.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radical and Rational Exponents
Radicals and rational exponents represent the same mathematical operation. For example, the nth root of a number can be written as that number raised to the power of 1/n. Understanding this equivalence allows simplification of expressions involving roots and fractional powers.
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Rational Exponents
Properties of Exponents
The properties of exponents, such as (a^m)^n = a^(m*n), are essential for simplifying expressions with powers raised to other powers. Applying these rules helps combine and reduce exponents efficiently, which is crucial in simplifying radical expressions.
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Reducing the Index of a Radical
Reducing the index of a radical involves rewriting the expression to have a smaller root index or converting it into a simpler form using exponent rules. This process often makes the expression easier to interpret or further simplify.
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Related Practice
Textbook Question
Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree.
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Textbook Question
Simplify the radical expressions in Exercises 67–74, if possible. ³√9⋅³√6
Textbook Question
Express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. -26 and -3
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Textbook Question
In Exercises 67–82, find each product. (7x2 y+1)(2x2 y−3)
Textbook Question
In Exercises 69–82, simplify each complex rational expression. (1/x + 1/y)/(x+y)