Solve for d square root of (50x6y3)/(9x8)=(5y6 square root of 2y)/(dx)

Question

Solvely Expert · Accepted Answer

Solution:

Step 1

Express the given equation in a simplified form: $\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{50 x^{6} y^{3}}{9 x^{8}}}$ .

Step 2

Simplify both sides of the equation.

Step 2.1

Reduce the fraction $\frac{50 x^{6} y^{3}}{9 x^{8}}$ by eliminating common factors.

Step 2.1.1

Extract $x^{6}$ from the numerator: $\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{x^{6} (50 y^{3})}{9 x^{8}}}$ .

Step 2.1.2

Extract $x^{6}$ from the denominator: $\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{x^{6} (50 y^{3})}{x^{6} (9 x^{2})}}$ .

Step 2.1.3

Cancel out $x^{6}$ : $\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{50 y^{3}}{9 x^{2}}}$ .

Step 2.1.4

Rewrite the simplified expression: $\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{50 y^{3}}{9 x^{2}}}$ .

Step 2.2

Express $\frac{50 y^{3}}{9 x^{2}}$ as ${(\frac{5 y}{3 x})}^{2} (2 y)$ .

Step 2.2.1

Factor out $(5 y)^{2}$ from $50 y^{3}$ : $\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{(5 y)^{2} \cdot 2 y}{9 x^{2}}}$ .

Step 2.2.2

Factor out $(3 x)^{2}$ from $9 x^{2}$ : $\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{(5 y)^{2} \cdot 2 y}{(3 x)^{2}}}$ .

Step 2.2.3

Reorganize the fraction: $\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{{(\frac{5 y}{3 x})}^{2} (2 y)}$ .

Step 2.3

Extract terms from under the radical: $\frac{5 y^{6} \sqrt{2 y}}{d x} = \frac{5 y}{3 x} \sqrt{2 y}$ .

Step 2.4

Combine $\frac{5 y}{3 x}$ and $\sqrt{2 y}$ : $\frac{5 y^{6} \sqrt{2 y}}{d x} = \frac{5 y \sqrt{2 y}}{3 x}$ .

Step 3

Cross-multiply the numerators and denominators: $5 y^{6} \sqrt{2 y} (3 x) = d x (5 y \sqrt{2 y})$ .

Step 4

Isolate $d$ to solve the equation.

Step 4.1

Rearrange the equation: $d x (5 y \sqrt{2 y}) = 5 y^{6} \sqrt{2 y} (3 x)$ .

Step 4.2

Simplify the equation.

Step 4.2.1

Remove parentheses: $d x (5 y \sqrt{2 y}) = 5 y^{6} \sqrt{2 y} (3 x)$ .

Step 4.2.2

Multiply $3$ by $5$ : $d x (5 y \sqrt{2 y}) = 15 y^{6} \sqrt{2 y} x$ .

Step 4.3

Divide both sides by $x (5 y \sqrt{2 y})$ and simplify.

Step 4.3.1

Divide the equation: $\frac{d x (5 y \sqrt{2 y})}{x (5 y \sqrt{2 y})} = \frac{15 y^{6} \sqrt{2 y} x}{x (5 y \sqrt{2 y})}$ .

Step 4.3.2

Simplify the left side by canceling common factors: $d = \frac{15 y^{6} \sqrt{2 y} x}{x (5 y \sqrt{2 y})}$ .

Step 4.3.3

Simplify the right side by canceling common factors: $d = \frac{3 y^{5} x}{x}$ .

Step 4.3.4

Cancel the common factor of $x$ : $d = 3 y^{5}$ .

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.

Solve for d square root of (50x^6y^3)/(9x^8)=(5y^6 square root of 2y)/(dx)

Answer