Algebra Examples

$\sqrt{\frac{50 x^{6} y^{3}}{9 x^{8}}} = \frac{5 y^{6} \sqrt{2 y}}{d x}$

Step 1

Rewrite the equation as $\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{50 x^{6} y^{3}}{9 x^{8}}}$ .

$\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{50 x^{6} y^{3}}{9 x^{8}}}$

Step 2

Simplify both sides.

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Step 2.1

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

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Step 2.1.1

Factor $x^{6}$ out of $50 x^{6} y^{3}$ .

$\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{x^{6} (50 y^{3})}{9 x^{8}}}$

Step 2.1.2

Factor $x^{6}$ out of $9 x^{8}$ .

$\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{x^{6} (50 y^{3})}{x^{6} (9 x^{2})}}$

Step 2.1.3

Cancel the common factor.

$\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{x^{6} (50 y^{3})}{x^{6} (9 x^{2})}}$

Step 2.1.4

Rewrite the expression.

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

Step 2.2

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

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Step 2.2.1

Factor the perfect power ${(5 y)}^{2}$ out of $50 y^{3}$ .

$\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{{(5 y)}^{2} \cdot (2 y)}{9 x^{2}}}$

Step 2.2.2

Factor the perfect power ${(3 x)}^{2}$ out of $9 x^{2}$ .

$\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{\frac{{(5 y)}^{2} \cdot (2 y)}{{(3 x)}^{2} \cdot 1}}$

Step 2.2.3

Rearrange the fraction $\frac{{(5 y)}^{2} \cdot (2 y)}{{(3 x)}^{2} \cdot 1}$ .

$\frac{5 y^{6} \sqrt{2 y}}{d x} = \sqrt{{(\frac{5 y}{3 x})}^{2} (2 y)}$

Step 2.3

Pull terms out 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

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

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

Step 4

Solve the equation for $d$ .

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Step 4.1

Rewrite the equation as $d x \cdot (5 y \sqrt{2 y}) = 5 y^{6} \sqrt{2 y} \cdot (3 x)$ .

$d x \cdot (5 y \sqrt{2 y}) = 5 y^{6} \sqrt{2 y} \cdot (3 x)$

Step 4.2

Simplify.

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Step 4.2.1

Remove parentheses.

$d x \cdot (5 y) \sqrt{2 y} = 5 y^{6} \sqrt{2 y} \cdot (3 x)$

Step 4.2.2

Remove parentheses.

$d x \cdot 5 y \sqrt{2 y} = 5 y^{6} \sqrt{2 y} \cdot (3 x)$

Step 4.2.3

Multiply $3$ by $5$ .

$d x \cdot 5 y \sqrt{2 y} = 15 y^{6} \sqrt{2 y} \cdot x$

Step 4.3

Divide each term in $d x \cdot (5 y \sqrt{2 y}) = 15 y^{6} \sqrt{2 y} \cdot x$ by $x \cdot 5 y \sqrt{2 y}$ and simplify.

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Step 4.3.1

Divide each term in $d x \cdot (5 y \sqrt{2 y}) = 15 y^{6} \sqrt{2 y} \cdot x$ by $x \cdot 5 y \sqrt{2 y}$ .

$\frac{d x \cdot (5 y \sqrt{2 y})}{x \cdot 5 y \sqrt{2 y}} = \frac{15 y^{6} \sqrt{2 y} \cdot x}{x \cdot 5 y \sqrt{2 y}}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $x$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{d x \cdot (5 y \sqrt{2 y})}{x \cdot 5 y \sqrt{2 y}} = \frac{15 y^{6} \sqrt{2 y} \cdot x}{x \cdot 5 y \sqrt{2 y}}$

Step 4.3.2.1.2

Rewrite the expression.

$\frac{d \cdot (5 y \sqrt{2 y})}{5 y \sqrt{2 y}} = \frac{15 y^{6} \sqrt{2 y} \cdot x}{x \cdot 5 y \sqrt{2 y}}$

Step 4.3.2.2

Cancel the common factor of $5$ .

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Step 4.3.2.2.1

Cancel the common factor.

$\frac{d \cdot (5 y \sqrt{2 y})}{5 y \sqrt{2 y}} = \frac{15 y^{6} \sqrt{2 y} \cdot x}{x \cdot 5 y \sqrt{2 y}}$

Step 4.3.2.2.2

Rewrite the expression.

$\frac{d \cdot (y \sqrt{2 y})}{y \sqrt{2 y}} = \frac{15 y^{6} \sqrt{2 y} \cdot x}{x \cdot 5 y \sqrt{2 y}}$

Step 4.3.2.3

Cancel the common factor of $y$ .

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Step 4.3.2.3.1

Cancel the common factor.

$\frac{d \cdot (y \sqrt{2 y})}{y \sqrt{2 y}} = \frac{15 y^{6} \sqrt{2 y} \cdot x}{x \cdot 5 y \sqrt{2 y}}$

Step 4.3.2.3.2

Rewrite the expression.

$\frac{d \cdot (\sqrt{2 y})}{\sqrt{2 y}} = \frac{15 y^{6} \sqrt{2 y} \cdot x}{x \cdot 5 y \sqrt{2 y}}$

Step 4.3.2.4

Cancel the common factor of $\sqrt{2 y}$ .

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Step 4.3.2.4.1

Cancel the common factor.

$\frac{d \cdot \sqrt{2 y}}{\sqrt{2 y}} = \frac{15 y^{6} \sqrt{2 y} \cdot x}{x \cdot 5 y \sqrt{2 y}}$

Step 4.3.2.4.2

Divide $d$ by $1$ .

$d = \frac{15 y^{6} \sqrt{2 y} \cdot x}{x \cdot 5 y \sqrt{2 y}}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Cancel the common factor of $15$ and $5$ .

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Step 4.3.3.1.1

Factor $5$ out of $15 y^{6} \sqrt{2 y} \cdot x$ .

$d = \frac{5 (3 y^{6} \sqrt{2 y} \cdot x)}{x \cdot 5 y \sqrt{2 y}}$

Step 4.3.3.1.2

Cancel the common factors.

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Step 4.3.3.1.2.1

Factor $5$ out of $x \cdot 5 y \sqrt{2 y}$ .

$d = \frac{5 (3 y^{6} \sqrt{2 y} \cdot x)}{5 (x y \sqrt{2 y})}$

Step 4.3.3.1.2.2

Cancel the common factor.

$d = \frac{5 (3 y^{6} \sqrt{2 y} \cdot x)}{5 (x y \sqrt{2 y})}$

Step 4.3.3.1.2.3

Rewrite the expression.

$d = \frac{3 y^{6} \sqrt{2 y} \cdot x}{x y \sqrt{2 y}}$

Step 4.3.3.2

Cancel the common factor of $y^{6}$ and $y$ .

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Step 4.3.3.2.1

Factor $y$ out of $3 y^{6} \sqrt{2 y} \cdot x$ .

$d = \frac{y (3 y^{5} \sqrt{2 y} \cdot x)}{x y \sqrt{2 y}}$

Step 4.3.3.2.2

Cancel the common factors.

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Step 4.3.3.2.2.1

Factor $y$ out of $x y \sqrt{2 y}$ .

$d = \frac{y (3 y^{5} \sqrt{2 y} \cdot x)}{y (x \sqrt{2 y})}$

Step 4.3.3.2.2.2

Cancel the common factor.

$d = \frac{y (3 y^{5} \sqrt{2 y} \cdot x)}{y (x \sqrt{2 y})}$

Step 4.3.3.2.2.3

Rewrite the expression.

$d = \frac{3 y^{5} \sqrt{2 y} \cdot x}{x \sqrt{2 y}}$

Step 4.3.3.3

Cancel the common factor of $\sqrt{2 y}$ .

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Step 4.3.3.3.1

Cancel the common factor.

$d = \frac{3 y^{5} \sqrt{2 y} \cdot x}{x \sqrt{2 y}}$

Step 4.3.3.3.2

Rewrite the expression.

$d = \frac{3 y^{5} \cdot x}{x}$

Step 4.3.3.4

Cancel the common factor of $x$ .

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Step 4.3.3.4.1

Cancel the common factor.

$d = \frac{3 y^{5} \cdot x}{x}$

Step 4.3.3.4.2

Divide $3 y^{5}$ by $1$ .

$d = 3 y^{5}$