Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Radical Expressions

College Algebra

0. Review of Algebra

Radical Expressions

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1:12 minutes

Problem 108

Textbook Question

In Exercises 101–108, simplify by reducing the index of the radical. ¹²√x^4y^8

Verified step by step guidance

insert step 1: Identify the given expression as a radical with an index of 12: \( \sqrt[12]{x^4y^8} \)

insert step 2: Recognize that simplifying a radical involves reducing the index by finding the greatest common factor (GCF) of the exponents and the index.

insert step 3: Determine the GCF of the exponents 4 and 8, which is 4.

insert step 4: Divide each exponent by the GCF: \( x^{4/4}y^{8/4} = x^1y^2 \).

insert step 5: Rewrite the expression with the reduced index: \( \sqrt[3]{x^1y^2} \).

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Key Concepts

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

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, represented by the radical symbol (√). The index of a radical indicates the degree of the root; for example, a square root has an index of 2, while a cube root has an index of 3. Understanding how to manipulate these expressions is crucial for simplification.

Properties of Exponents

The properties of exponents govern how to simplify expressions involving powers. Key rules include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the power of a product ( (ab)^n = a^n * b^n). These rules are essential for reducing the index of radicals and simplifying expressions.

Simplifying Radicals

Simplifying radicals involves expressing a radical in its simplest form, which often includes reducing the index. This process may require factoring the radicand (the expression inside the radical) into perfect squares or cubes, allowing for the extraction of whole numbers from the radical. Mastery of this concept is key to solving problems involving radicals efficiently.