Basic Math Examples

$\sqrt{9 r^{7} s^{5}} + 9 r s {(\sqrt{r^{3} s})}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $9 r^{7} s^{5}$ as ${(3 r^{3} s^{2})}^{2} (r s)$ .

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Step 1.1.1

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} r^{7} s^{5}} + 9 r s {(\sqrt{r^{3} s})}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.1.2

Factor out $r^{6}$ .

$\sqrt{3^{2} (r^{6} r) s^{5}} + 9 r s {(\sqrt{r^{3} s})}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.1.3

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

$\sqrt{3^{2} ({(r^{3})}^{2} r) s^{5}} + 9 r s {(\sqrt{r^{3} s})}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.1.4

Factor out $s^{4}$ .

$\sqrt{3^{2} ({(r^{3})}^{2} r) (s^{4} s)} + 9 r s {(\sqrt{r^{3} s})}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.1.5

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

$\sqrt{3^{2} ({(r^{3})}^{2} r) ({(s^{2})}^{2} s)} + 9 r s {(\sqrt{r^{3} s})}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.1.6

Move $r$ .

$\sqrt{3^{2} ({(r^{3})}^{2}) {(s^{2})}^{2} r s} + 9 r s {(\sqrt{r^{3} s})}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.1.7

Rewrite $3^{2} ({(r^{3})}^{2}) {(s^{2})}^{2}$ as ${(3 r^{3} s^{2})}^{2}$ .

$\sqrt{{(3 r^{3} s^{2})}^{2} r s} + 9 r s {(\sqrt{r^{3} s})}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.1.8

Add parentheses.

$\sqrt{{(3 r^{3} s^{2})}^{2} (r s)} + 9 r s {(\sqrt{r^{3} s})}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.2

Pull terms out from under the radical.

$3 r^{3} s^{2} \sqrt{r s} + 9 r s {(\sqrt{r^{3} s})}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.3

Rewrite $r^{3} s$ as $r^{2} (r s)$ .

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Step 1.3.1

Factor out $r^{2}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r s {\sqrt{r^{2} r s}}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.3.2

Add parentheses.

$3 r^{3} s^{2} \sqrt{r s} + 9 r s {\sqrt{r^{2} (r s)}}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.4

Pull terms out from under the radical.

$3 r^{3} s^{2} \sqrt{r s} + 9 r s {(r \sqrt{r s})}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.5

Apply the product rule to $r \sqrt{r s}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r s (r^{5} {\sqrt{r s}}^{5}) - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.6

Multiply $r$ by $r^{5}$ by adding the exponents.

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Step 1.6.1

Move $r^{5}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 (r^{5} r) s {\sqrt{r s}}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.6.2

Multiply $r^{5}$ by $r$ .

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Step 1.6.2.1

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

$3 r^{3} s^{2} \sqrt{r s} + 9 (r^{5} r^{1}) s {\sqrt{r s}}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.6.2.2

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

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{5 + 1} s {\sqrt{r s}}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.6.3

Add $5$ and $1$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{6} s {\sqrt{r s}}^{5} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.7

Rewrite ${\sqrt{r s}}^{5}$ as $\sqrt{{(r s)}^{5}}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{6} s \sqrt{{(r s)}^{5}} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.8

Apply the product rule to $r s$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{6} s \sqrt{r^{5} s^{5}} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.9

Rewrite $r^{5} s^{5}$ as ${(r^{2} s^{2})}^{2} (r s)$ .

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Step 1.9.1

Factor out $r^{4}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{6} s \sqrt{r^{4} r s^{5}} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.9.2

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

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{6} s \sqrt{{(r^{2})}^{2} r s^{5}} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.9.3

Factor out $s^{4}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{6} s \sqrt{{(r^{2})}^{2} r (s^{4} s)} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.9.4

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

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{6} s \sqrt{{(r^{2})}^{2} r ({(s^{2})}^{2} s)} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.9.5

Move $r$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{6} s \sqrt{{(r^{2})}^{2} {(s^{2})}^{2} r s} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.9.6

Rewrite ${(r^{2})}^{2} {(s^{2})}^{2}$ as ${(r^{2} s^{2})}^{2}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{6} s \sqrt{{(r^{2} s^{2})}^{2} r s} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.9.7

Add parentheses.

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{6} s \sqrt{{(r^{2} s^{2})}^{2} (r s)} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.10

Pull terms out from under the radical.

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{6} s (r^{2} s^{2} \sqrt{r s}) - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.11

Multiply $r^{6}$ by $r^{2}$ by adding the exponents.

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Step 1.11.1

Move $r^{2}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 (r^{2} r^{6}) s (s^{2} \sqrt{r s}) - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.11.2

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

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{2 + 6} s (s^{2} \sqrt{r s}) - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.11.3

Add $2$ and $6$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s (s^{2} \sqrt{r s}) - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.12

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

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Step 1.12.1

Move $s^{2}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} (s^{2} s) \sqrt{r s} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.12.2

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

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Step 1.12.2.1

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

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} (s^{2} s^{1}) \sqrt{r s} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.12.2.2

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

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{2 + 1} \sqrt{r s} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.12.3

Add $2$ and $1$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r s \sqrt{r^{5} s^{3}}$

Step 1.13

Rewrite $r^{5} s^{3}$ as ${(r^{2} s)}^{2} (r s)$ .

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Step 1.13.1

Factor out $r^{4}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r s \sqrt{r^{4} r s^{3}}$

Step 1.13.2

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

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r s \sqrt{{(r^{2})}^{2} r s^{3}}$

Step 1.13.3

Factor out $s^{2}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r s \sqrt{{(r^{2})}^{2} r (s^{2} s)}$

Step 1.13.4

Move $r$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r s \sqrt{{(r^{2})}^{2} s^{2} r s}$

Step 1.13.5

Rewrite ${(r^{2})}^{2} s^{2}$ as ${(r^{2} s)}^{2}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r s \sqrt{{(r^{2} s)}^{2} r s}$

Step 1.13.6

Add parentheses.

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r s \sqrt{{(r^{2} s)}^{2} (r s)}$

Step 1.14

Pull terms out from under the radical.

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r s (r^{2} s \sqrt{r s})$

Step 1.15

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

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Step 1.15.1

Move $r^{2}$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 (r^{2} r) s (s \sqrt{r s})$

Step 1.15.2

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

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Step 1.15.2.1

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

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 (r^{2} r^{1}) s (s \sqrt{r s})$

Step 1.15.2.2

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

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r^{2 + 1} s (s \sqrt{r s})$

Step 1.15.3

Add $2$ and $1$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r^{3} s (s \sqrt{r s})$

Step 1.16

Multiply $s$ by $s$ by adding the exponents.

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Step 1.16.1

Move $s$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r^{3} (s \cdot s) \sqrt{r s}$

Step 1.16.2

Multiply $s$ by $s$ .

$3 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s} - 9 r^{3} s^{2} \sqrt{r s}$

Step 2

Subtract $9 r^{3} s^{2} \sqrt{r s}$ from $3 r^{3} s^{2} \sqrt{r s}$ .

$- 6 r^{3} s^{2} \sqrt{r s} + 9 r^{8} s^{3} \sqrt{r s}$