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 53

Chapter 1, Problem 53

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ∜ m² • ∜ m²

Verified step by step guidance

Recognize that the expression involves the product of two fourth roots: $\sqrt[4]{m^{2}} \cdot \sqrt[4]{m^{2}}$.

Use the property of radicals that states $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$ to combine the two radicals into one: $\sqrt[4]{m^{2} \cdot m^{2}}$.

Multiply the expressions inside the radical: $m^{2} \cdot m^{2} = m^{2+2} = m^{4}$, so the expression becomes $\sqrt[4]{m^{4}}$.

Rewrite the radical expression as an exponent: $\sqrt[4]{m^{4}} = \left(m^{4}\right)^{\frac{1}{4}}$.

Apply the power of a power rule by multiplying the exponents: $m^{4 \cdot \frac{1}{4}} = m^{1}$, which simplifies to $m$.

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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, such as square roots or fourth roots. Understanding how to manipulate radicals, including simplifying and combining them, is essential. Key properties include that the product of two radicals with the same index equals the radical of the product.

Multiplication of Radicals with the Same Index

When multiplying radicals that have the same root index, you can multiply the radicands (the expressions inside the radicals) together under a single radical. For example, ∜a • ∜b = ∜(a•b). This simplifies the expression and is crucial for solving the given problem.

Exponents and Radicals Relationship

Radicals can be expressed as fractional exponents, where the nth root of a number is the same as raising that number to the power of 1/n. For example, ∜m² = m^(2/4) = m^(1/2). This relationship helps in simplifying and combining radical expressions involving variables.

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