Perform the indicated operation of multiplication or division on the rational expressions and simplify.
$\frac{27}{7 m} \cdot \frac{m}{24}$
Answer (2)
Multiply the numerators and denominators: 7 m 27 ⋅ 24 m = 168 m 27 m .
Cancel the common factor 'm': 168 m 27 m = 168 27 .
Simplify the numerical coefficients by dividing both by their greatest common divisor, which is 3: 168 27 = 56 9 .
The simplified expression is: 56 9 .
Explanation
Understanding the Problem We are given the expression 7 m 27 ⋅ 24 m and asked to simplify it. This involves multiplying the two rational expressions and reducing the result to its simplest form.
Multiplying the Expressions To multiply the rational expressions, we multiply the numerators and the denominators: 7 m 27 ⋅ 24 m = 7 m ⋅ 24 27 m = 168 m 27 m
Canceling Common Factors Now, we simplify the resulting fraction. First, we can cancel the common factor of 'm' from the numerator and the denominator, since m = 0 : 168 m 27 m = 168 27
Simplifying Numerical Coefficients Next, we simplify the numerical coefficients. We find the greatest common divisor (GCD) of 27 and 168. From the previous calculation, we know that GCD(27, 24) = 3. Since 24 is a factor of 168 ( 168 = 7 ⋅ 24 ), 3 must also be a factor of 168. So, we divide both 27 and 168 by 3: 168 27 = 168 ÷ 3 27 ÷ 3 = 56 9
Final Answer The simplified expression is 56 9 .
Examples
Rational expressions are useful in various real-world scenarios, such as calculating rates, proportions, and scaling recipes. For example, if you need to adjust the amount of ingredients in a recipe based on the number of servings, you might use rational expressions to maintain the correct ratios. Simplifying these expressions helps in making quick and accurate calculations, ensuring the recipe turns out as expected. This skill is also applicable in fields like engineering and finance, where ratios and proportions are frequently used.
The expression 7 m 27 ⋅ 24 m simplifies to 56 9 after multiplying, canceling the common factor 'm', and reducing the resulting fraction. The greatest common divisor of 27 and 168 is used to achieve the final simplification. Thus, the final simplified result is 56 9 .
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