Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 51

Chapter 1, Problem 51

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ⁵√x² • ⁵√x³

Verified step by step guidance

Identify the operation: You are multiplying two fifth roots, \(\sqrt[5]{x^2}\) and \(\sqrt[5]{x^3}\).

Recall the property of radicals that allows multiplication under the same root: \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\).

Apply this property to combine the radicals: \(\sqrt[5]{x^2} \cdot \sqrt[5]{x^3} = \sqrt[5]{x^2 \cdot x^3}\).

Use the laws of exponents to multiply inside the radical: \(x^2 \cdot x^3 = x^{2+3} = x^5\).

Rewrite the expression as \(\sqrt[5]{x^5}\) and recognize that the fifth root of \(x^5\) simplifies to \(x\) because \(\sqrt[5]{x^5} = x^{5/5} = x\).

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Key Concepts

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

Properties of Radicals

Radicals represent roots of numbers or expressions. The nth root of a product equals the product of the nth roots, allowing multiplication under a single radical. For example, ⁿ√a × ⁿ√b = ⁿ√(a × b), which simplifies operations involving radicals with the same index.

Multiplication of Radicals with the Same Index

When multiplying radicals that have the same root index, you can combine them into one radical by multiplying their radicands. For instance, ⁵√x² × ⁵√x³ = ⁵√(x² × x³) = ⁵√x⁵, simplifying the expression efficiently.

Simplifying Radicals Using Exponent Rules

Radicals can be expressed as fractional exponents, where ⁿ√x^m = x^(m/n). This allows simplification by applying exponent rules, such as adding exponents when multiplying like bases, making it easier to simplify expressions like ⁵√x⁵ = x^(5/5) = x.

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Let A = {2, 4, 6, 8, 10, 12}, B = {2, 4, 8, 10}, C = {4, 10, 12}, D = {2, 10}, andU = {2, 4, 6, 8, 10, 12, 14}. Determine whether each statement is true or false. A ⊆ U

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