Simplify (3y)/(2 square root of 3xy^3-5x^3)

$\frac{3 y}{2 \sqrt{3 x y^{3}} - 5 x^{3}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x y^{3}}$ as ${(3 x y^{3})}^{\frac{1}{2}}$ .

$\frac{3 y}{2 {(3 x y^{3})}^{\frac{1}{2}} - 5 x^{3}}$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $3 x y^{3}$ .

$\frac{3 y}{2 ({(3 x)}^{\frac{1}{2}} {(y^{3})}^{\frac{1}{2}}) - 5 x^{3}}$

Apply the product rule to $3 x$ .

$\frac{3 y}{2 (3^{\frac{1}{2}} x^{\frac{1}{2}} {(y^{3})}^{\frac{1}{2}}) - 5 x^{3}}$

Step 3

Multiply the exponents in ${(y^{3})}^{\frac{1}{2}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 y}{2 (3^{\frac{1}{2}} x^{\frac{1}{2}} y^{3 (\frac{1}{2})}) - 5 x^{3}}$

Combine $3$ and $\frac{1}{2}$ .

$\frac{3 y}{2 (3^{\frac{1}{2}} x^{\frac{1}{2}} y^{\frac{3}{2}}) - 5 x^{3}}$

Step 4

Factor $x^{\frac{1}{2}}$ out of $2 \cdot 3^{\frac{1}{2}} x^{\frac{1}{2}} y^{\frac{3}{2}} - 5 x^{3}$ .

Factor $x^{\frac{1}{2}}$ out of $2 \cdot 3^{\frac{1}{2}} x^{\frac{1}{2}} y^{\frac{3}{2}}$ .

$\frac{3 y}{x^{\frac{1}{2}} (2 \cdot 3^{\frac{1}{2}} y^{\frac{3}{2}}) - 5 x^{3}}$

Factor $x^{\frac{1}{2}}$ out of $- 5 x^{3}$ .

$\frac{3 y}{x^{\frac{1}{2}} (2 \cdot 3^{\frac{1}{2}} y^{\frac{3}{2}}) + x^{\frac{1}{2}} (- 5 x^{\frac{5}{2}})}$

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (2 \cdot 3^{\frac{1}{2}} y^{\frac{3}{2}}) + x^{\frac{1}{2}} (- 5 x^{\frac{5}{2}})$ .

$\frac{3 y}{x^{\frac{1}{2}} (2 \cdot 3^{\frac{1}{2}} y^{\frac{3}{2}} - 5 x^{\frac{5}{2}})}$