Calculus Examples

$(\sqrt{245 x^{5} y^{7}}) \cdot (\sqrt{10 x y^{6}})$

Step 1

Rewrite $245 x^{5} y^{7}$ as ${(7 x^{2} y^{3})}^{2} \cdot (5 x y)$ .

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Step 1.1

Factor $49$ out of $245$ .

$\sqrt{49 (5) x^{5} y^{7}} \cdot \sqrt{10 x y^{6}}$

Step 1.2

Rewrite $49$ as $7^{2}$ .

$\sqrt{7^{2} \cdot 5 x^{5} y^{7}} \cdot \sqrt{10 x y^{6}}$

Step 1.3

Factor out $x^{4}$ .

$\sqrt{7^{2} \cdot 5 (x^{4} x) y^{7}} \cdot \sqrt{10 x y^{6}}$

Step 1.4

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

$\sqrt{7^{2} \cdot 5 ({(x^{2})}^{2} x) y^{7}} \cdot \sqrt{10 x y^{6}}$

Step 1.5

Factor out $y^{6}$ .

$\sqrt{7^{2} \cdot 5 ({(x^{2})}^{2} x) (y^{6} y)} \cdot \sqrt{10 x y^{6}}$

Step 1.6

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

$\sqrt{7^{2} \cdot 5 ({(x^{2})}^{2} x) ({(y^{3})}^{2} y)} \cdot \sqrt{10 x y^{6}}$

Step 1.7

Move $x$ .

$\sqrt{7^{2} \cdot 5 ({(x^{2})}^{2}) {(y^{3})}^{2} x y} \cdot \sqrt{10 x y^{6}}$

Step 1.8

Move $5$ .

$\sqrt{(7^{2} ({(x^{2})}^{2})) {(y^{3})}^{2} \cdot 5 x y} \cdot \sqrt{10 x y^{6}}$

Step 1.9

Rewrite $(7^{2} ({(x^{2})}^{2})) {(y^{3})}^{2}$ as ${(7 x^{2} y^{3})}^{2}$ .

$\sqrt{{(7 x^{2} y^{3})}^{2} \cdot 5 x y} \cdot \sqrt{10 x y^{6}}$

Step 1.10

Add parentheses.

$\sqrt{{(7 x^{2} y^{3})}^{2} \cdot 5 (x y)} \cdot \sqrt{10 x y^{6}}$

Step 1.11

Add parentheses.

$\sqrt{{(7 x^{2} y^{3})}^{2} \cdot (5 x y)} \cdot \sqrt{10 x y^{6}}$

Step 2

Pull terms out from under the radical.

$7 x^{2} y^{3} \sqrt{5 x y} \cdot \sqrt{10 x y^{6}}$

Step 3

Rewrite $10 x y^{6}$ as ${(y^{3})}^{2} \cdot (10 x)$ .

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Step 3.1

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

$7 x^{2} y^{3} \sqrt{5 x y} \cdot \sqrt{10 x {(y^{3})}^{2}}$

Step 3.2

Move $x$ .

$7 x^{2} y^{3} \sqrt{5 x y} \cdot \sqrt{10 {(y^{3})}^{2} x}$

Step 3.3

Reorder $10$ and ${(y^{3})}^{2}$ .

$7 x^{2} y^{3} \sqrt{5 x y} \cdot \sqrt{{(y^{3})}^{2} \cdot 10 x}$

Step 3.4

Add parentheses.

$7 x^{2} y^{3} \sqrt{5 x y} \cdot \sqrt{{(y^{3})}^{2} \cdot (10 x)}$

Step 4

Pull terms out from under the radical.

$7 x^{2} y^{3} \sqrt{5 x y} \cdot (y^{3} \sqrt{10 x})$

Step 5

Multiply $y^{3}$ by $y^{3}$ by adding the exponents.

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Step 5.1

Move $y^{3}$ .

$7 x^{2} (y^{3} y^{3}) \sqrt{5 x y} \cdot \sqrt{10 x}$

Step 5.2

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

$7 x^{2} y^{3 + 3} \sqrt{5 x y} \cdot \sqrt{10 x}$

Step 5.3

Add $3$ and $3$ .

$7 x^{2} y^{6} \sqrt{5 x y} \cdot \sqrt{10 x}$

Step 6

Multiply $7 x^{2} y^{6} \sqrt{5 x y} \sqrt{10 x}$ .

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Step 6.1

Combine using the product rule for radicals.

$7 x^{2} y^{6} \sqrt{10 x (5 x y)}$

Step 6.2

Multiply $5$ by $10$ .

$7 x^{2} y^{6} \sqrt{50 x (x y)}$

Step 6.3

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

$7 x^{2} y^{6} \sqrt{50 (x^{1} x) y}$

Step 6.4

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

$7 x^{2} y^{6} \sqrt{50 (x^{1} x^{1}) y}$

Step 6.5

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

$7 x^{2} y^{6} \sqrt{50 x^{1 + 1} y}$

Step 6.6

Add $1$ and $1$ .

$7 x^{2} y^{6} \sqrt{50 x^{2} y}$

Step 7

Rewrite $50 x^{2} y$ as ${(5 x)}^{2} \cdot (2 y)$ .

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Step 7.1

Factor $25$ out of $50$ .

$7 x^{2} y^{6} \sqrt{25 (2) x^{2} y}$

Step 7.2

Rewrite $25$ as $5^{2}$ .

$7 x^{2} y^{6} \sqrt{5^{2} \cdot 2 x^{2} y}$

Step 7.3

Move $2$ .

$7 x^{2} y^{6} \sqrt{5^{2} x^{2} \cdot 2 y}$

Step 7.4

Rewrite $5^{2} x^{2}$ as ${(5 x)}^{2}$ .

$7 x^{2} y^{6} \sqrt{{(5 x)}^{2} \cdot 2 y}$

Step 7.5

Add parentheses.

$7 x^{2} y^{6} \sqrt{{(5 x)}^{2} \cdot (2 y)}$

Step 8

Pull terms out from under the radical.

$7 x^{2} y^{6} (5 x \sqrt{2 y})$

Step 9

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

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Step 9.1

Move $x$ .

$7 (x \cdot x^{2}) y^{6} (5 \sqrt{2 y})$

Step 9.2

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

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Step 9.2.1

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

$7 (x^{1} x^{2}) y^{6} (5 \sqrt{2 y})$

Step 9.2.2

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

$7 x^{1 + 2} y^{6} (5 \sqrt{2 y})$

Step 9.3

Add $1$ and $2$ .

$7 x^{3} y^{6} (5 \sqrt{2 y})$

Step 10

Multiply $5$ by $7$ .

$35 x^{3} y^{6} \sqrt{2 y}$