Basic Math Examples

$\sqrt{4 z^{7} y^{5}} + 9 z^{2} \sqrt{z^{3} y^{5}} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $4 z^{7} y^{5}$ as ${(2 z^{3} y^{2})}^{2} (z y)$ .

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Step 1.1.1

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} z^{7} y^{5}} + 9 z^{2} \sqrt{z^{3} y^{5}} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.1.2

Factor out $z^{6}$ .

$\sqrt{2^{2} (z^{6} z) y^{5}} + 9 z^{2} \sqrt{z^{3} y^{5}} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.1.3

Rewrite $z^{6}$ as ${(z^{3})}^{2}$ .

$\sqrt{2^{2} ({(z^{3})}^{2} z) y^{5}} + 9 z^{2} \sqrt{z^{3} y^{5}} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.1.4

Factor out $y^{4}$ .

$\sqrt{2^{2} ({(z^{3})}^{2} z) (y^{4} y)} + 9 z^{2} \sqrt{z^{3} y^{5}} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.1.5

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

$\sqrt{2^{2} ({(z^{3})}^{2} z) ({(y^{2})}^{2} y)} + 9 z^{2} \sqrt{z^{3} y^{5}} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.1.6

Move $z$ .

$\sqrt{2^{2} ({(z^{3})}^{2}) {(y^{2})}^{2} z y} + 9 z^{2} \sqrt{z^{3} y^{5}} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.1.7

Rewrite $2^{2} ({(z^{3})}^{2}) {(y^{2})}^{2}$ as ${(2 z^{3} y^{2})}^{2}$ .

$\sqrt{{(2 z^{3} y^{2})}^{2} z y} + 9 z^{2} \sqrt{z^{3} y^{5}} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.1.8

Add parentheses.

$\sqrt{{(2 z^{3} y^{2})}^{2} (z y)} + 9 z^{2} \sqrt{z^{3} y^{5}} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.2

Pull terms out from under the radical.

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{2} \sqrt{z^{3} y^{5}} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.3

Rewrite $z^{3} y^{5}$ as ${(z y^{2})}^{2} (z y)$ .

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Step 1.3.1

Factor out $z^{2}$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{2} \sqrt{z^{2} z y^{5}} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.3.2

Factor out $y^{4}$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{2} \sqrt{z^{2} z (y^{4} y)} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.3.3

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{2} \sqrt{z^{2} z ({(y^{2})}^{2} y)} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.3.4

Move $z$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{2} \sqrt{z^{2} {(y^{2})}^{2} z y} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.3.5

Rewrite $z^{2} {(y^{2})}^{2}$ as ${(z y^{2})}^{2}$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{2} \sqrt{{(z y^{2})}^{2} z y} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.3.6

Add parentheses.

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{2} \sqrt{{(z y^{2})}^{2} (z y)} - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.4

Pull terms out from under the radical.

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{2} (z y^{2} \sqrt{z y}) - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.5

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

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Step 1.5.1

Move $z$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 (z \cdot z^{2}) (y^{2} \sqrt{z y}) - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.5.2

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

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Step 1.5.2.1

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

$2 z^{3} y^{2} \sqrt{z y} + 9 (z^{1} z^{2}) (y^{2} \sqrt{z y}) - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.5.2.2

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

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{1 + 2} (y^{2} \sqrt{z y}) - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.5.3

Add $1$ and $2$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z y \sqrt{z^{5} y^{3}}$

Step 1.6

Rewrite $z^{5} y^{3}$ as ${(z^{2} y)}^{2} (z y)$ .

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Step 1.6.1

Factor out $z^{4}$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z y \sqrt{z^{4} z y^{3}}$

Step 1.6.2

Rewrite $z^{4}$ as ${(z^{2})}^{2}$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z y \sqrt{{(z^{2})}^{2} z y^{3}}$

Step 1.6.3

Factor out $y^{2}$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z y \sqrt{{(z^{2})}^{2} z (y^{2} y)}$

Step 1.6.4

Move $z$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z y \sqrt{{(z^{2})}^{2} y^{2} z y}$

Step 1.6.5

Rewrite ${(z^{2})}^{2} y^{2}$ as ${(z^{2} y)}^{2}$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z y \sqrt{{(z^{2} y)}^{2} z y}$

Step 1.6.6

Add parentheses.

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z y \sqrt{{(z^{2} y)}^{2} (z y)}$

Step 1.7

Pull terms out from under the radical.

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z y (z^{2} y \sqrt{z y})$

Step 1.8

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

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Step 1.8.1

Move $z^{2}$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 (z^{2} z) y (y \sqrt{z y})$

Step 1.8.2

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

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Step 1.8.2.1

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

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 (z^{2} z^{1}) y (y \sqrt{z y})$

Step 1.8.2.2

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

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z^{2 + 1} y (y \sqrt{z y})$

Step 1.8.3

Add $2$ and $1$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z^{3} y (y \sqrt{z y})$

Step 1.9

Multiply $y$ by $y$ by adding the exponents.

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Step 1.9.1

Move $y$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z^{3} (y \cdot y) \sqrt{z y}$

Step 1.9.2

Multiply $y$ by $y$ .

$2 z^{3} y^{2} \sqrt{z y} + 9 z^{3} (y^{2} \sqrt{z y}) - 5 z^{3} y^{2} \sqrt{z y}$

Step 2

Simplify by adding terms.

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Step 2.1

Move $z^{3}$ .

$2 y^{2} z^{3} \sqrt{y z} + 9 y^{2} z^{3} \sqrt{y z} - 5 z^{3} y^{2} \sqrt{z y}$

Step 2.2

Add $2 y^{2} z^{3} \sqrt{y z}$ and $9 y^{2} z^{3} \sqrt{y z}$ .

$11 y^{2} z^{3} \sqrt{y z} - 5 z^{3} y^{2} \sqrt{z y}$

Step 3

Subtract $5 z^{3} y^{2} \sqrt{z y}$ from $11 y^{2} z^{3} \sqrt{y z}$ .

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Step 3.1

Move $z^{3}$ .

$11 y^{2} z^{3} \sqrt{y z} - 5 y^{2} z^{3} \sqrt{y z}$

Step 3.2

Subtract $5 y^{2} z^{3} \sqrt{y z}$ from $11 y^{2} z^{3} \sqrt{y z}$ .

$6 y^{2} z^{3} \sqrt{y z}$