Solve for d square root of (50x^6y^3)/(9x^8)=(5y^6 square root of 2y)/(dx)
The problem provided is a mathematical equation that involves solving for the variable 'd'. The equation includes a radical expression on both sides, specifically the square root of a rational expression involving powers of x and y on the left, and a product of a monomial, the square root of a variable, and another variable on the right. The task is to manipulate the equation to isolate 'd' and express it in terms of the other variables and constants present in the equation. This would likely require knowledge of algebraic operations involving roots and rational expressions, as well as the properties of exponents.
$\sqrt{\frac{50 x^{6} y^{3}}{9 x^{8}}} = \frac{5 y^{6} \sqrt{2 y}}{d x}$
Answer
Solution:
Step 1
Express the given equation in a simplified form: $\frac{5y^6\sqrt{2y}}{dx} = \sqrt{\frac{50x^6y^3}{9x^8}}$.
Step 2
Simplify both sides of the equation.
Step 2.1
Reduce the fraction $\frac{50x^6y^3}{9x^8}$ by eliminating common factors.
Step 2.1.1
Extract $x^6$ from the numerator: $\frac{5y^6\sqrt{2y}}{dx} = \sqrt{\frac{x^6(50y^3)}{9x^8}}$.
Step 2.1.2
Extract $x^6$ from the denominator: $\frac{5y^6\sqrt{2y}}{dx} = \sqrt{\frac{x^6(50y^3)}{x^6(9x^2)}}$.
Step 2.1.3
Cancel out $x^6$: $\frac{5y^6\sqrt{2y}}{dx} = \sqrt{\frac{50y^3}{9x^2}}$.
Step 2.1.4
Rewrite the simplified expression: $\frac{5y^6\sqrt{2y}}{dx} = \sqrt{\frac{50y^3}{9x^2}}$.
Step 2.2
Express $\frac{50y^3}{9x^2}$ as $\left(\frac{5y}{3x}\right)^2(2y)$.
Step 2.2.1
Factor out $(5y)^2$ from $50y^3$: $\frac{5y^6\sqrt{2y}}{dx} = \sqrt{\frac{(5y)^2 \cdot 2y}{9x^2}}$.
Step 2.2.2
Factor out $(3x)^2$ from $9x^2$: $\frac{5y^6\sqrt{2y}}{dx} = \sqrt{\frac{(5y)^2 \cdot 2y}{(3x)^2}}$.
Step 2.2.3
Reorganize the fraction: $\frac{5y^6\sqrt{2y}}{dx} = \sqrt{\left(\frac{5y}{3x}\right)^2(2y)}$.
Step 2.3
Extract terms from under the radical: $\frac{5y^6\sqrt{2y}}{dx} = \frac{5y}{3x}\sqrt{2y}$.
Step 2.4
Combine $\frac{5y}{3x}$ and $\sqrt{2y}$: $\frac{5y^6\sqrt{2y}}{dx} = \frac{5y\sqrt{2y}}{3x}$.
Step 3
Cross-multiply the numerators and denominators: $5y^6\sqrt{2y}(3x) = dx(5y\sqrt{2y})$.
Step 4
Isolate $d$ to solve the equation.
Step 4.1
Rearrange the equation: $dx(5y\sqrt{2y}) = 5y^6\sqrt{2y}(3x)$.
Step 4.2
Simplify the equation.
Step 4.2.1
Remove parentheses: $dx(5y\sqrt{2y}) = 5y^6\sqrt{2y}(3x)$.
Step 4.2.2
Multiply $3$ by $5$: $dx(5y\sqrt{2y}) = 15y^6\sqrt{2y}x$.
Step 4.3
Divide both sides by $x(5y\sqrt{2y})$ and simplify.
Step 4.3.1
Divide the equation: $\frac{dx(5y\sqrt{2y})}{x(5y\sqrt{2y})} = \frac{15y^6\sqrt{2y}x}{x(5y\sqrt{2y})}$.
Step 4.3.2
Simplify the left side by canceling common factors: $d = \frac{15y^6\sqrt{2y}x}{x(5y\sqrt{2y})}$.
Step 4.3.3
Simplify the right side by canceling common factors: $d = \frac{3y^5x}{x}$.
Step 4.3.4
Cancel the common factor of $x$: $d = 3y^5$.
Knowledge Notes:
The problem-solving process involves simplifying algebraic expressions and solving equations. The key knowledge points include:
Radical Expressions: Understanding how to simplify radical expressions, including square roots, and how to extract terms from under the radical.
Fraction Simplification: Reducing fractions by canceling common factors in the numerator and denominator.
Cross-Multiplication: A technique used to solve proportions by multiplying the numerator of one fraction by the denominator of the other fraction and setting the products equal to each other.
Algebraic Manipulation: Rearranging equations, factoring, and canceling like terms to isolate the variable of interest.
LaTeX Formatting: Representing mathematical expressions using LaTeX syntax to ensure clarity and readability in written solutions.
