Textbook Question
Write a decimal number that has 5 in the thousands place, 0 in the tenths place, and 4 in the ten-thousandths place.
Ch. R - Review of Basic Concepts
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 1, Problem 80a
Simplify each complex fraction. [ y + 1/(y2-9) ] / [ 1/(y + 3) ]
Verified step by step guidance1
Identify the complex fraction: \(\frac{y + \frac{1}{y^2 - 9}}{\frac{1}{y + 3}}\).
Recognize that \(y^2 - 9\) is a difference of squares and factor it as \(y^2 - 9 = (y - 3)(y + 3)\).
Rewrite the numerator by combining the terms over a common denominator: \(y + \frac{1}{(y - 3)(y + 3)} = \frac{y(y - 3)(y + 3)}{(y - 3)(y + 3)} + \frac{1}{(y - 3)(y + 3)}\).
Combine the fractions in the numerator: \(\frac{y(y - 3)(y + 3) + 1}{(y - 3)(y + 3)}\).
Divide the numerator by the denominator fraction by multiplying by the reciprocal: \(\frac{y(y - 3)(y + 3) + 1}{(y - 3)(y + 3)} \times (y + 3)\), then simplify the expression by canceling common factors.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Fractions
A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression to eliminate the smaller fractions, often by finding a common denominator or multiplying numerator and denominator by the least common denominator (LCD).
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Complex Conjugates
Factoring Polynomials
Factoring involves expressing a polynomial as a product of simpler polynomials. Recognizing special products like the difference of squares, e.g., y^2 - 9 = (y - 3)(y + 3), helps simplify expressions and cancel common factors in fractions.
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07:30Introduction to Factoring Polynomials
Division of Fractions
Dividing by a fraction is equivalent to multiplying by its reciprocal. To simplify a complex fraction, rewrite the division as multiplication by flipping the denominator fraction, then multiply numerators and denominators accordingly.
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05:45Radical Expressions with Fractions
Related Practice
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Simplify each complex fraction.