Perform the indicated operation of multiplication or division on the rational expressions and simplify.
$\frac{25 a^2}{24 b^2} \div \frac{5 b}{4 a^2}$
Answer (2)
Rewrite the division as multiplication by the reciprocal: 24 b 2 25 a 2 × 5 b 4 a 2 .
Multiply the numerators and denominators: 24 b 2 × 5 b 25 a 2 × 4 a 2 = 120 b 3 100 a 4 .
Simplify the fraction by dividing by the greatest common divisor (20): 120 b 3 100 a 4 = 6 b 3 5 a 4 .
The simplified expression is: 6 b 3 5 a 4 .
Explanation
Understanding the Problem We are given the expression 24 b 2 25 a 2 ÷ 4 a 2 5 b . Our goal is to perform the division and simplify the resulting expression.
Rewriting as Multiplication To divide rational expressions, we multiply by the reciprocal of the second fraction. So, we rewrite the division as multiplication: 24 b 2 25 a 2 ÷ 4 a 2 5 b = 24 b 2 25 a 2 × 5 b 4 a 2
Multiplying Numerators and Denominators Now, we multiply the numerators and the denominators: 24 b 2 × 5 b 25 a 2 × 4 a 2 = 120 b 3 100 a 4
Simplifying the Expression Next, we simplify the expression by canceling common factors. We can divide both the numerator and the denominator by their greatest common divisor, which is 20: 120 b 3 100 a 4 = 120 ÷ 20 100 ÷ 20 × b 3 a 4 = 6 b 3 5 a 4
Final Answer Therefore, the simplified expression is 6 b 3 5 a 4 .
Examples
Rational expressions are useful in various real-world applications, such as calculating rates, proportions, and scaling quantities. For instance, if you are mixing a solution that requires a specific ratio of two ingredients, rational expressions can help you determine the correct amounts to use. Suppose you need to scale a recipe that calls for 3 2 cup of flour to 4 1 cup of sugar. Using rational expressions, you can easily adjust the quantities to maintain the correct ratio, ensuring the recipe turns out as expected. This concept extends to many fields, including chemistry, engineering, and finance, where proportional relationships are essential.
To solve 24 b 2 25 a 2 ÷ 4 a 2 5 b , we multiply by the reciprocal and simplify the result. This gives us the simplified expression 6 b 3 5 a 4 . The key steps include rewriting the division as multiplication, multiplying, and then simplifying.
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