Simplify (3y)/(2x^3)+(5z)/(8xy^2)

$\frac{3 y}{2 x^{3}} + \frac{5 z}{8 x y^{2}}$

Step 1

To write $\frac{3 y}{2 x^{3}}$ as a fraction with a common denominator, multiply by $\frac{4 y^{2}}{4 y^{2}}$ .

$\frac{3 y}{2 x^{3}} \cdot \frac{4 y^{2}}{4 y^{2}} + \frac{5 z}{8 x y^{2}}$

Step 2

To write $\frac{5 z}{8 x y^{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{3 y}{2 x^{3}} \cdot \frac{4 y^{2}}{4 y^{2}} + \frac{5 z}{8 x y^{2}} \cdot \frac{x^{2}}{x^{2}}$

Step 3

Write each expression with a common denominator of $8 x^{3} y^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 y}{2 x^{3}}$ by $\frac{4 y^{2}}{4 y^{2}}$ .

$\frac{3 y (4 y^{2})}{2 x^{3} (4 y^{2})} + \frac{5 z}{8 x y^{2}} \cdot \frac{x^{2}}{x^{2}}$

Multiply $4$ by $2$ .

$\frac{3 y (4 y^{2})}{8 x^{3} y^{2}} + \frac{5 z}{8 x y^{2}} \cdot \frac{x^{2}}{x^{2}}$

Multiply $\frac{5 z}{8 x y^{2}}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{3 y (4 y^{2})}{8 x^{3} y^{2}} + \frac{5 z x^{2}}{8 x y^{2} x^{2}}$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$\frac{3 y (4 y^{2})}{8 x^{3} y^{2}} + \frac{5 z x^{2}}{8 (x^{2} x) y^{2}}$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$\frac{3 y (4 y^{2})}{8 x^{3} y^{2}} + \frac{5 z x^{2}}{8 (x^{2} x^{1}) y^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 y (4 y^{2})}{8 x^{3} y^{2}} + \frac{5 z x^{2}}{8 x^{2 + 1} y^{2}}$

Add $2$ and $1$ .

$\frac{3 y (4 y^{2})}{8 x^{3} y^{2}} + \frac{5 z x^{2}}{8 x^{3} y^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 y (4 y^{2}) + 5 z x^{2}}{8 x^{3} y^{2}}$

Step 5

Simplify the numerator.

Rewrite using the commutative property of multiplication.

$\frac{3 \cdot 4 y \cdot y^{2} + 5 z x^{2}}{8 x^{3} y^{2}}$

Multiply $y$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$\frac{3 \cdot 4 (y^{2} y) + 5 z x^{2}}{8 x^{3} y^{2}}$

Multiply $y^{2}$ by $y$ .

Raise $y$ to the power of $1$ .

$\frac{3 \cdot 4 (y^{2} y^{1}) + 5 z x^{2}}{8 x^{3} y^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 \cdot 4 y^{2 + 1} + 5 z x^{2}}{8 x^{3} y^{2}}$

Add $2$ and $1$ .

$\frac{3 \cdot 4 y^{3} + 5 z x^{2}}{8 x^{3} y^{2}}$

Multiply $3$ by $4$ .

$\frac{12 y^{3} + 5 z x^{2}}{8 x^{3} y^{2}}$