Basic Math Examples

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

Step 1

Combine.

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

Step 2

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

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

Step 3

Cancel the common factors.

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Step 3.1

Factor $z^{3}$ out of $y^{4} z^{3}$ .

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Cancel the common factors.

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Step 5.1

Factor $y$ out of $y^{4}$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

$\frac{(z^{2} (z^{- 2} + 5 z)) (y^{\frac{3}{2}} - 2 y)}{y^{3}}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $z^{- 2}$ out of $z^{- 2} + 5 z$ .

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Step 6.1.1

Multiply by $1$ .

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

Step 6.1.2

Factor $z^{- 2}$ out of $5 z$ .

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

Step 6.1.3

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

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

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

Step 6.2

Combine exponents.

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Step 6.2.1

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

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Step 6.2.1.1

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

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

Step 6.2.1.2

Subtract $2$ from $2$ .

$\frac{z^{0} (1 + 5 z^{3}) (y^{\frac{3}{2}} - 2 y)}{y^{3}}$

Step 6.2.2

Simplify $z^{0}$ .

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

Step 6.2.3

Multiply $1 + 5 z^{3}$ by $1$ .

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

Step 6.3

Factor $y$ out of $y^{\frac{3}{2}} - 2 y$ .

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Step 6.3.1

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

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

Step 6.3.2

Factor $y$ out of $- 2 y$ .

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

Step 6.3.3

Factor $y$ out of $y \cdot y^{\frac{1}{2}} + y \cdot - 2$ .

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

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

Step 7

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

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

Step 8

Cancel the common factors.

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Step 8.1

Factor $y$ out of $y^{3}$ .

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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