Textbook Question
In Exercises 87–106, perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. (2.4×10^−2)/(4.8×10^-6)
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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 102
Simplify by reducing the index of the radical.
Verified step by step guidance1
Identify the expression inside the radical: the fourth root of \(7^2\), which can be written as \(\sqrt[4]{7^2}\).
Recall the property of radicals and exponents: \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\). Apply this to rewrite the expression as \(7^{\frac{2}{4}}\).
Simplify the fraction in the exponent: \(\frac{2}{4} = \frac{1}{2}\), so the expression becomes \(7^{\frac{1}{2}}\).
Recognize that \(7^{\frac{1}{2}}\) is equivalent to the square root of 7, or \(\sqrt{7}\).
Thus, the simplified form of \(\sqrt[4]{7^2}\) is \(\sqrt{7}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radical Expressions and Indices
A radical expression involves roots, such as square roots or fourth roots, indicated by an index. The index shows the degree of the root, for example, the fourth root (⁴√) means the number raised to the 1/4 power. Understanding how to interpret and manipulate these indices is essential for simplifying radicals.
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Radical Expressions with Fractions
Reducing the Index of a Radical
Reducing the index means rewriting a radical with a smaller root index or converting it into an equivalent expression with fractional exponents. This often involves expressing the radicand with powers that allow simplification, such as rewriting ⁴√(7²) as 7^(2/4) and then simplifying the fraction.
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Fractional Exponents and Simplification
Fractional exponents represent roots and powers simultaneously, where a^(m/n) equals the nth root of a raised to the mth power. Simplifying expressions with fractional exponents involves reducing the fraction and rewriting the expression in simplest form, which helps in reducing the index of radicals effectively.
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Related Practice
Textbook Question
In Exercises 103–110, insert either <, >, or = in the shaded area to make a true statement. |−20| □ |−50|
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Textbook Question
Factor completely, or state that the polynomial is prime. 3x^4-12x^2
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Textbook Question
In Exercises 103–114, factor completely. 6x4+35x2−6
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Textbook Question
Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. 282,000,000,000/0.00141
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Textbook Question
Factor and simplify each algebraic expression.
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