Find the values of $a$ and $b$ that make the second expression equivalent to the first expression. Assume that $x>0$ and $y \geq 0$. $\sqrt{\frac{126 x y^5}{32 x^3}}=\sqrt{\frac{63 y^5}{a x^b}}$ $a=$ $\square$ and $b=$ $\square$
Answer (1)
Simplify the left side of the equation: 32 x 3 126 x y 5 = 16 x 2 63 y 5 .
Compare the simplified expression with the right side: 16 x 2 63 y 5 = a x b 63 y 5 .
Equate the denominators: 16 x 2 = a x b .
Determine the values of a and b : a = 16 , b = 2 .
Explanation
Problem Analysis We are given the equation 32 x 3 126 x y 5 = a x b 63 y 5 and we want to find the values of a and b that make the two expressions equivalent. We are also given that 0"> x > 0 and y ≥ 0 .
Simplifying the Left Side First, let's simplify the left side of the equation. We have 32 x 3 126 x y 5 We can simplify the fraction 32 126 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us 32 126 = 16 63 .
Also, we can simplify x 3 x to x 2 1 .
So, the expression becomes 16 x 2 63 y 5
Comparing Expressions Now, we compare this simplified expression with the right side of the equation, which is a x b 63 y 5 .
We have 16 x 2 63 y 5 = a x b 63 y 5 For the two expressions to be equal, the denominators must be equal. Therefore, we must have 16 x 2 = a x b
Finding a and b From the equation 16 x 2 = a x b , we can deduce that a = 16 and b = 2 .
Final Answer Therefore, the values of a and b that make the two expressions equivalent are a = 16 and b = 2 .
Examples
Imagine you are comparing the performance of two different algorithms in terms of their time complexity. The expressions given in the problem could represent the time complexity of these algorithms, where x is the input size and y is some other parameter. By simplifying and comparing these expressions, you can determine the values of a and b that make the two algorithms have equivalent time complexities. This helps in understanding the efficiency and scalability of the algorithms.
