Algebra Examples

$\frac{3 z}{2 x y^{3}} - \frac{2 x}{4 x^{3} z^{2}}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $2$ and $4$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{3 z}{2 x y^{3}} - \frac{2 (x)}{4 x^{3} z^{2}}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $4 x^{3} z^{2}$ .

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

Step 1.1.2.2

Cancel the common factor.

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

Step 1.1.2.3

Rewrite the expression.

$\frac{3 z}{2 x y^{3}} - \frac{x}{2 x^{3} z^{2}}$

Step 1.2

Cancel the common factor of $x$ and $x^{3}$ .

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Step 1.2.1

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

$\frac{3 z}{2 x y^{3}} - \frac{x^{1}}{2 x^{3} z^{2}}$

Step 1.2.2

Factor $x$ out of $x^{1}$ .

$\frac{3 z}{2 x y^{3}} - \frac{x \cdot 1}{2 x^{3} z^{2}}$

Step 1.2.3

Cancel the common factors.

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Step 1.2.3.1

Factor $x$ out of $2 x^{3} z^{2}$ .

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

Step 1.2.3.2

Cancel the common factor.

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

Step 1.2.3.3

Rewrite the expression.

$\frac{3 z}{2 x y^{3}} - \frac{1}{2 x^{2} z^{2}}$

Step 2

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

$\frac{3 z}{2 x y^{3}} \cdot \frac{x z^{2}}{x z^{2}} - \frac{1}{2 x^{2} z^{2}}$

Step 3

To write $- \frac{1}{2 x^{2} z^{2}}$ as a fraction with a common denominator, multiply by $\frac{y^{3}}{y^{3}}$ .

$\frac{3 z}{2 x y^{3}} \cdot \frac{x z^{2}}{x z^{2}} - \frac{1}{2 x^{2} z^{2}} \cdot \frac{y^{3}}{y^{3}}$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Multiply $\frac{1}{2 x^{2} z^{2}}$ by $\frac{y^{3}}{y^{3}}$ .

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

Step 4.7

Reorder the factors of $2 x^{2} y^{3} z^{2}$ .

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

Step 4.8

Reorder the factors of $2 x^{2} z^{2} y^{3}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

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

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Step 6.1

Move $z^{2}$ .

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

$\frac{3 z^{2 + 1} x - y^{3}}{2 y^{3} x^{2} z^{2}}$

Step 6.3

Add $2$ and $1$ .

$\frac{3 z^{3} x - y^{3}}{2 y^{3} x^{2} z^{2}}$