Problem 19 Reduce each rational expression ... [FREE SOLUTION]

Chapter 6: Problem 19

Reduce each rational expression to its lowest terms. $$\frac{6}{57}$$

Short Answer

Expert verified

$ \frac{2}{19} $

Step by step solution

Identify the Greatest Common Divisor (GCD)

Determine the Greatest Common Divisor (GCD) of the numerator (6) and the denominator (57). The GCD is the largest number that divides both 6 and 57 evenly.

Factorize the Numbers

List the factors of 6 and 57. The factors of 6 are 1, 2, 3, and 6. The factors of 57 are 1, 3, 19, and 57. The common factors are 1 and 3. Therefore, the GCD is 3.

Divide Numerator and Denominator by GCD

Divide both the numerator and the denominator by their GCD (3). This gives: \[ \frac{6 \text{梅} 3}{57 \text{梅} 3} = \frac{2}{19} \]

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Key Concepts

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

Greatest Common Divisor (GCD)

To simplify a rational expression, you first need to find the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD is the largest number that can exactly divide both the numerator and the denominator. Finding the GCD is essential because it helps you reduce the fraction to its simplest form. For the fraction $\frac{6}{57}$, we start by determining the GCD of 6 and 57. We list out the factors of both numbers: 6 has factors of 1, 2, 3, and 6, while 57 has factors of 1, 3, 19, and 57. The largest common factor between these numbers is 3, making it the GCD.

Factoring

Factoring is a crucial step when simplifying rational expressions. When you factor numbers, you break them down into their prime components. This makes it easier to find common factors.
Let's understand it with an example. The number 6 can be factorized into 2 and 3, i.e., 6 = 2 x 3. Similarly, 57 can be factorized into 3 and 19, i.e., 57 = 3 x 19.
By listing out the factors of both the numerator and the denominator, you can easily identify the GCD, which is the number you need to divide both by to simplify the fraction.

Reducing Fractions

Reducing fractions means simplifying them to their lowest terms. Once you know the GCD, you can accomplish this by dividing both the numerator and the denominator by the GCD. For $\frac{6}{57}$, we found that the GCD is 3. By dividing the numerator (6) by 3, we get 2.
Similarly, by dividing the denominator (57) by 3, we get 19. Thus, the fraction $\frac{6}{57}$ is simplified to $\frac{2}{19}$. This makes the fraction easier to understand and work with in further calculations.
Always ensure that your final fraction cannot be reduced any further by checking for any common factors.

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91影视

Reduce each rational expression to its lowest terms. $$\frac{6}{57}$$

Short Answer

Step by step solution

Identify the Greatest Common Divisor (GCD)

Factorize the Numbers

Divide Numerator and Denominator by GCD

Key Concepts

Greatest Common Divisor (GCD)

Factoring

Reducing Fractions

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