Textbook Question
Find the term indicated in each expansion. (x2 + y)22; the term containing y14
Ch. 8 - Sequences, Induction, and Probability
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 9, Problem 47
Express each repeating decimal as a fraction in lowest terms.
Verified step by step guidance1
Let \( x = 0.474747\ldots \) where the digits "47" repeat indefinitely.
Since the repeating block has 2 digits, multiply \( x \) by \( 10^{2} = 100 \) to shift the decimal point two places to the right: \( 100x = 47.474747\ldots \).
Set up the equation by subtracting the original \( x \) from this new expression to eliminate the repeating part: \( 100x - x = 47.474747\ldots - 0.474747\ldots \).
Simplify the subtraction to get \( 99x = 47 \), because the repeating decimals cancel out.
Solve for \( x \) by dividing both sides by 99: \( x = \frac{47}{99} \). Then, check if the fraction can be simplified to lowest terms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Repeating Decimals
A repeating decimal is a decimal number in which a digit or group of digits repeats infinitely. For example, 0.474747... has '47' repeating endlessly. Understanding the pattern of repetition is essential to convert such decimals into fractions.
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Geometric Series
A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. In the context of repeating decimals, the infinite sum of the repeating parts forms a geometric series, which can be summed using a formula to find an exact fraction.
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Fraction Simplification
After converting a repeating decimal into a fraction, it is important to simplify the fraction to its lowest terms by dividing numerator and denominator by their greatest common divisor. This ensures the fraction is expressed in the simplest and most standard form.
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Related Practice
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