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Chapter 6: Problem 22

Reduce each rational expression to its lowest terms. $$\frac{242}{154}$$

Short Answer

Expert verified
\( \frac{11}{7} \)

Step by step solution

01

- Find the GCD

Identify the greatest common divisor (GCD) of both the numerator (242) and the denominator (154). The GCD of 242 and 154 is 2.
02

- Divide the Numerator and Denominator by GCD

Divide both the numerator and the denominator by the GCD, 2.\(\frac{242 \, \div \, 2}{154 \, \div \, 2} = \frac{121}{77}\)
03

- Simplify Further if Necessary

Check if the resulting fraction \( \frac{121}{77} \) can be simplified further. The GCD of 121 and 77 is 11. Divide both by 11.\(\frac{121 \, \div \, 11}{77 \, \div \, 11} = \frac{11}{7}\)

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Key Concepts

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

Greatest Common Divisor (GCD)
The greatest common divisor (GCD) is a key concept in simplifying rational expressions. It represents the largest number that can evenly divide both the numerator and the denominator.

To find the GCD of two numbers, you can use different methods like listing factors, the Euclidean algorithm, or prime factorization. For example:
  • **Listing Factors**: Write down all the factors of each number and find the highest common one.
  • **Euclidean Algorithm**: This involves a series of division steps to find the GCD.
  • **Prime Factorization**: Break down each number to its prime factors and multiply the common primes.

Using the GCD makes it easier to divide and simplify fractions.
Divide Fractions
Dividing fractions includes simplifying each part (numerator and denominator) by their GCD. For example, in the fraction \(\frac{242}{154}\), the GCD is found to be 2 at first.

By dividing both parts by 2, you simplify to \(\frac{121}{77}\).
This process removes any common factors and reduces the fraction step-by-step.

To further ensure simplicity, check the new fraction:\(\frac{121}{77}\). Using the GCD again, this time which is 11, simplifies the fraction to \(\frac{11}{7}\).
This method shows how efficiency in division helps break down complex fractions into simpler ones.
Lowest Terms
Putting a fraction in its lowest terms means simplifying it so that the numerator and denominator have no common divisors other than 1.

For instance, starting with \(\frac{242}{154}\), we follow these steps:
  • **Find GCD**: First, 2, dividing gives \(\frac{121}{77}\).
  • **Divide Again**: Check GCD (which is now 11), divide to get \(\frac{11}{7}\).
  • **Result**: \(\frac{11}{7}\) has no further common factors, making it in the lowest terms.

Achieving the lowest terms ensures the fraction is in its simplest business. Always recheck the final fraction for any common factors to confirm completeness.

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Most popular questions from this chapter

Solve each problem. Aspect ratio. The aspect ratio for a television screen is the ratio of the width to the height of the screen. For a high definition TV it is 16 to \(9 .\) What is the height of a screen that is 48 inches wide?

Find the solution set to each equation. $$\frac{x}{6}=\frac{5}{x-1}$$

Solve each equation. Identify each equation as a conditional equation, an inconsistent equation, or an identity. State the solution sets to the identities using interval notation. $$\frac{1}{x}+\frac{1}{x^{2}}=\frac{x+2}{x^{2}}$$

Perform the indicated operations. Variables in exponents represent integers. $$\frac{w^{2 b}+2 w^{b}-8}{w^{2 b}+3 w^{b}-4} \div \frac{w^{2 b}-w^{b}-2}{w^{2 b}-1}$$

Perform the indicated operations. $$\frac{\left(-w y^{2}\right)^{3}}{3 w^{2} y} \cdot \frac{(2 w y)^{2}}{4 w y^{3}}$$
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