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Chapter 4: Problem 81

Reduce the given fraction to lowest terms. $$\frac{-12 y^{5}}{98 y^{6}}$$

Short Answer

Expert verified
\(\frac{-6}{49y}\)

Step by step solution

01

Identify the Common Factor

Start by identifying any common factors shared by the numerator and denominator. In this case, both the numerator (-12) and the denominator (98) are divisible by 2.
02

Divide by the Common Factor

Divide the numerator and the denominator by the greatest common factor, which is 2. This gives us:\[\frac{-12 \div 2}{98 \div 2} = \frac{-6}{49}\]
03

Simplify the Variables

The variable terms in the fraction are \(y^5\) in the numerator and \(y^6\) in the denominator. According to the laws of exponents, subtract the exponents when dividing like bases:\[\frac{y^5}{y^6} = y^{5-6} = y^{-1}\]This simplifies to \(\frac{1}{y}\).
04

Combine the Results

Combine the simplified numeric and variable terms to get the fully reduced fraction:\[\frac{-6}{49} \times \frac{1}{y} = \frac{-6}{49y}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Common Factors
In mathematics, a common factor is a number that divides two or more numbers without leaving a remainder. When you're tasked with reducing a fraction to its lowest terms, finding the common factors is crucial. For example, in the given problem, both -12 and 98 have common factors. We first identify the smallest positive common factor, which is 2.
  • Identify numbers that equally divide both the numerator and the denominator.
  • In this case, 2 is the number that divides both -12 and 98 without leaving any remainder.
This commonality helps us in simplifying fractions effectively.
The Laws of Exponents
Exponents provide a shorthand way of writing repeated multiplication. The laws of exponents help simplify expressions involving the same base. In the problem, we encounter variable terms like \(y^5\) and \(y^6\).
  • When dividing two expressions with the same base, you subtract the exponents: \(a^m / a^n = a^{m-n}\).
  • For \(y^5 / y^6\), subtracting the exponents gives us \(y^{5-6}\) or \(y^{-1}\).
  • An exponent of -1 implies the reciprocal: \(y^{-1} = 1/y\).
Understanding these rules can greatly ease the process of simplifying algebraic fractions.
Simplifying Fractions
Simplifying fractions involves both numeric and variable components in the given expression. The process of reducing fractions combines the elimination of common factors and the application of exponent laws.
  • First, factor the numerator and denominator by their greatest common factor, observed earlier as 2.
  • Next, apply exponent laws to the variable parts to reduce \(y^5/y^6\) to \(1/y\).
  • Finally, recombine the simplified numeric and variable parts to acquire the fully reduced fraction, which yields \(\frac{-6}{49y}\).
By following these steps in sequence, you can reduce fractions to their simplest form efficiently.

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Most popular questions from this chapter

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