Basic Math Examples

$\sqrt{15 r^{3}} \cdot \sqrt{3 r^{3}}$

Step 1

Rewrite $15 r^{3}$ as $r^{2} \cdot (15 r)$ .

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Step 1.1

Factor out $r^{2}$ .

$\sqrt{15 (r^{2} r)} \cdot \sqrt{3 r^{3}}$

Step 1.2

Reorder $15$ and $r^{2}$ .

$\sqrt{r^{2} \cdot 15 r} \cdot \sqrt{3 r^{3}}$

Step 1.3

Add parentheses.

$\sqrt{r^{2} \cdot (15 r)} \cdot \sqrt{3 r^{3}}$

$\sqrt{r^{2} \cdot (15 r)} \cdot \sqrt{3 r^{3}}$

Step 2

Pull terms out from under the radical.

$r \sqrt{15 r} \cdot \sqrt{3 r^{3}}$

Step 3

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

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Step 3.1

Factor out $r^{2}$ .

$r \sqrt{15 r} \cdot \sqrt{3 (r^{2} r)}$

Step 3.2

Reorder $3$ and $r^{2}$ .

$r \sqrt{15 r} \cdot \sqrt{r^{2} \cdot 3 r}$

Step 3.3

Add parentheses.

$r \sqrt{15 r} \cdot \sqrt{r^{2} \cdot (3 r)}$

$r \sqrt{15 r} \cdot \sqrt{r^{2} \cdot (3 r)}$

Step 4

Pull terms out from under the radical.

$r \sqrt{15 r} \cdot (r \sqrt{3 r})$

Step 5

Multiply $r$ by $r$ by adding the exponents.

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Step 5.1

Move $r$ .

$r \cdot r \sqrt{15 r} \cdot \sqrt{3 r}$

Step 5.2

Multiply $r$ by $r$ .

$r^{2} \sqrt{15 r} \cdot \sqrt{3 r}$

$r^{2} \sqrt{15 r} \cdot \sqrt{3 r}$

Step 6

Multiply $r^{2} \sqrt{15 r} \sqrt{3 r}$ .

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Step 6.1

Combine using the product rule for radicals.

$r^{2} \sqrt{3 r (15 r)}$

Step 6.2

Multiply $15$ by $3$ .

$r^{2} \sqrt{45 r \cdot r}$

Step 6.3

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

$r^{2} \sqrt{45 (r^{1} r)}$

Step 6.4

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

$r^{2} \sqrt{45 (r^{1} r^{1})}$

Step 6.5

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

$r^{2} \sqrt{45 r^{1 + 1}}$

Step 6.6

Add $1$ and $1$ .

$r^{2} \sqrt{45 r^{2}}$

$r^{2} \sqrt{45 r^{2}}$

Step 7

Rewrite $45 r^{2}$ as ${(3 r)}^{2} \cdot 5$ .

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Step 7.1

Factor $9$ out of $45$ .

$r^{2} \sqrt{9 (5) r^{2}}$

Step 7.2

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

$r^{2} \sqrt{3^{2} \cdot 5 r^{2}}$

Step 7.3

Move $5$ .

$r^{2} \sqrt{3^{2} r^{2} \cdot 5}$

Step 7.4

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

$r^{2} \sqrt{{(3 r)}^{2} \cdot 5}$

$r^{2} \sqrt{{(3 r)}^{2} \cdot 5}$

Step 8

Pull terms out from under the radical.

$r^{2} (3 r \sqrt{5})$

Step 9

Rewrite using the commutative property of multiplication.

$3 \sqrt{5} r^{2} r$

Step 10

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

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Step 10.1

Move $r$ .

$3 \sqrt{5} (r \cdot r^{2})$

Step 10.2

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

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Step 10.2.1

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

$3 \sqrt{5} (r^{1} r^{2})$

Step 10.2.2

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

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

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

Step 10.3

Add $1$ and $2$ .

$3 \sqrt{5} r^{3}$

$3 \sqrt{5} r^{3}$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$