$\frac{24 x \sqrt{2-x}}{34 y}$

To simplify the expression 34 y 24 x 2 − x ​ ​ , we reduce it to 17 y 12 x 2 − x ​ ​ and find that x ≤ 2 and y  = 0 are the necessary restrictions. This ensures that the values used in the expression are valid and maintain defined operations. The simplification and valid ranges help prevent any nonsensical solutions in mathematical applications.
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Answered by Anonymous | 2025-07-04

Simplify the fraction 34 24 ​ to 17 12 ​ .
Rewrite the expression as 17 y 12 x 2 − x ​ ​ .
Determine the restriction on x : 2 − x ≥ 0 , which implies x ≤ 2 .
State the restriction on y : y  = 0 .
The simplified expression with restrictions is: 17 y 12 x 2 − x ​ ​ ​ , where x ≤ 2 and y  = 0 .

Explanation

Understanding the Expression We are given the expression 34 y 24 x 2 − x ​ ​ Our goal is to analyze and simplify this expression, identifying any restrictions on the variables involved.

Simplifying the Fraction First, we simplify the fraction 34 24 ​ by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us: 34 24 ​ = 34 ÷ 2 24 ÷ 2 ​ = 17 12 ​ So, the expression becomes: 17 y 12 x 2 − x ​ ​

Identifying Restrictions Next, we need to consider the restrictions on the variables x and y . Since we have a square root, the expression inside the square root must be non-negative: 2 − x ≥ 0 Adding x to both sides, we get: 2 ≥ x So, x ≤ 2 . Also, since y is in the denominator, it cannot be zero: y  = 0

Final Result and Restrictions Therefore, the simplified expression is 17 y 12 x 2 − x ​ ​ with the restrictions x ≤ 2 and y  = 0 .


Examples
Consider a scenario where you are calculating the flow rate of a liquid through a pipe. The expression might represent a simplified model where 'x' relates to the liquid's viscosity and 'y' to the pipe's diameter. The square root term could account for turbulence effects. Understanding such expressions and their variable restrictions ensures your calculations remain physically meaningful and prevent nonsensical results, like negative viscosity or zero diameter.

Answered by GinnyAnswer | 2025-07-04