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 28

Chapter 1, Problem 28

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent positive real numbers. p^5/4

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Identify the given expression: $p^{5/4}$. This is in exponential form with a fractional exponent.

Recall the rule that a fractional exponent $a^{m/n}$ can be rewritten in radical form as $\sqrt[n]{a^m}$ or equivalently $\left(\sqrt[n]{a}\right)^m$.

Apply this rule to rewrite $p^{5/4}$ as $\sqrt[4]{p^5}$, which means the fourth root of $p$ raised to the fifth power.

Alternatively, express $p^{5/4}$ as $\left(\sqrt[4]{p}\right)^5$, which means the fourth root of $p$ raised to the fifth power.

Since $p$ is a positive real number, this radical form is valid. Evaluating further depends on the value of $p$, so leave the expression in radical form as $\sqrt[4]{p^5}$ or $\left(\sqrt[4]{p}\right)^5$.

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Key Concepts

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

Exponential and Radical Forms

Exponential form expresses numbers using exponents, such as p^(5/4). Radical form uses roots, like the fourth root of p raised to the fifth power. Understanding how to convert between these forms is essential for simplifying and evaluating expressions.

Fractional Exponents

A fractional exponent a/b means taking the b-th root of the base raised to the a-th power, i.e., p^(a/b) = (b√p)^a. This concept links exponents and radicals, allowing expressions to be rewritten and evaluated in different forms.

Properties of Positive Real Numbers

Assuming variables represent positive real numbers ensures that roots and exponents are well-defined and real-valued. This avoids complications with negative bases or complex numbers when converting between radical and exponential forms.

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Simplify each expression. Assume all variables represent nonzero real numbers. -(2x⁰y⁴)³

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Multiply or divide as indicated. Write answers in lowest terms as needed. (3/20)∙(5/21)

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Add or subtract, as indicated. $- (8 x^{3} + x - 3) + (2 x^{3} + x^{2}) - (4 x^{2} + 3 x - 1)$

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Insert ∈ or ∉ in each blank to make the resulting statement true. 6 _____ {3, 4, 5, 6}

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Textbook Question

Concept Check When directed to completely factor the polynomial $4 x^{2} y^{5} - 8 x y^{3}$ ,a student wrote $2 x y^{3} (2 x y^{2} - 4)$ . When the teacher did not give him full credit, he complained because when his answer is multiplied out, the result is the original polynomial. Give the correct answer.

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Textbook Question

Write each rational expression in lowest terms. m² - 4m + 4 / m² + m - 6

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent positive real numbers. p5/4

Verified video answer for a similar problem:

Key Concepts

Exponential and Radical Forms

Fractional Exponents

Properties of Positive Real Numbers

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent positive real numbers. p^5/4