Algebra Examples

$\frac{\sqrt[3]{8 x^{6} y^{12}}}{\sqrt{4 x^{4} y^{8}}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = \sqrt[3]{8 x^{6} y^{12}}$ and $g (y) = \sqrt{4 x^{4} y^{8}}$ .

$\frac{\sqrt{4 x^{4} y^{8}} \frac{d}{d y} [\sqrt[3]{8 x^{6} y^{12}}] - \sqrt[3]{8 x^{6} y^{12}} \frac{d}{d y} [\sqrt{4 x^{4} y^{8}}]}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 2

Rewrite $\frac{d}{d y} [\sqrt[3]{8 x^{6} y^{12}}]$ as $\frac{d}{d y} [2 x^{2} y^{4}]$ .

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Step 2.1

Rewrite $8 x^{6} y^{12}$ as ${(2 x^{2} y^{4})}^{3}$ .

$\frac{\sqrt{4 x^{4} y^{8}} \frac{d}{d y} [\sqrt[3]{{(2 x^{2} y^{4})}^{3}}] - \sqrt[3]{8 x^{6} y^{12}} \frac{d}{d y} [\sqrt{4 x^{4} y^{8}}]}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 2.2

Pull terms out from under the radical, assuming real numbers.

$\frac{\sqrt{4 x^{4} y^{8}} \frac{d}{d y} [2 x^{2} y^{4}] - \sqrt[3]{8 x^{6} y^{12}} \frac{d}{d y} [\sqrt{4 x^{4} y^{8}}]}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 3

Since $2 x^{2}$ is constant with respect to $y$ , the derivative of $2 x^{2} y^{4}$ with respect to $y$ is $2 x^{2} \frac{d}{d y} [y^{4}]$ .

$\frac{\sqrt{4 x^{4} y^{8}} (2 x^{2} \frac{d}{d y} [y^{4}]) - \sqrt[3]{8 x^{6} y^{12}} \frac{d}{d y} [\sqrt{4 x^{4} y^{8}}]}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 4$ .

$\frac{\sqrt{4 x^{4} y^{8}} (2 x^{2} (4 y^{3})) - \sqrt[3]{8 x^{6} y^{12}} \frac{d}{d y} [\sqrt{4 x^{4} y^{8}}]}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 5

Simplify the expression.

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Step 5.1

Multiply $4$ by $2$ .

$\frac{\sqrt{4 x^{4} y^{8}} (8 x^{2} y^{3}) - \sqrt[3]{8 x^{6} y^{12}} \frac{d}{d y} [\sqrt{4 x^{4} y^{8}}]}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 5.2

Reorder the factors of $8 x^{2} y^{3}$ .

$\frac{\sqrt{4 x^{4} y^{8}} (8 y^{3} x^{2}) - \sqrt[3]{8 x^{6} y^{12}} \frac{d}{d y} [\sqrt{4 x^{4} y^{8}}]}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 6

Rewrite $\frac{d}{d y} [\sqrt{4 x^{4} y^{8}}]$ as $\frac{d}{d y} [2 x^{2} y^{4}]$ .

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Step 6.1

Rewrite $4 x^{4} y^{8}$ as ${(2 x^{2} y^{4})}^{2}$ .

$\frac{\sqrt{4 x^{4} y^{8}} (8 y^{3} x^{2}) - \sqrt[3]{8 x^{6} y^{12}} \frac{d}{d y} [\sqrt{{(2 x^{2} y^{4})}^{2}}]}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 6.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{4 x^{4} y^{8}} (8 y^{3} x^{2}) - \sqrt[3]{8 x^{6} y^{12}} \frac{d}{d y} [2 x^{2} y^{4}]}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 7

Since $2 x^{2}$ is constant with respect to $y$ , the derivative of $2 x^{2} y^{4}$ with respect to $y$ is $2 x^{2} \frac{d}{d y} [y^{4}]$ .

$\frac{\sqrt{4 x^{4} y^{8}} (8 y^{3} x^{2}) - \sqrt[3]{8 x^{6} y^{12}} (2 x^{2} \frac{d}{d y} [y^{4}])}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 8

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 4$ .

$\frac{\sqrt{4 x^{4} y^{8}} (8 y^{3} x^{2}) - \sqrt[3]{8 x^{6} y^{12}} (2 x^{2} (4 y^{3}))}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 9

Simplify the expression.

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Step 9.1

Multiply $4$ by $2$ .

$\frac{\sqrt{4 x^{4} y^{8}} (8 y^{3} x^{2}) - \sqrt[3]{8 x^{6} y^{12}} (8 x^{2} y^{3})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 9.2

Reorder the factors of $8 x^{2} y^{3}$ .

$\frac{\sqrt{4 x^{4} y^{8}} (8 y^{3} x^{2}) - \sqrt[3]{8 x^{6} y^{12}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10

Simplify.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Simplify each term.

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Step 10.1.1.1

Rewrite using the commutative property of multiplication.

$\frac{8 \sqrt{4 x^{4} y^{8}} (y^{3} x^{2}) - \sqrt[3]{8 x^{6} y^{12}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.2

Rewrite $4 x^{4} y^{8}$ as ${(2 x^{2} y^{4})}^{2}$ .

$\frac{8 \sqrt{{(2 x^{2} y^{4})}^{2}} (y^{3} x^{2}) - \sqrt[3]{8 x^{6} y^{12}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{8 (2 x^{2} y^{4}) (y^{3} x^{2}) - \sqrt[3]{8 x^{6} y^{12}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.4

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

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Step 10.1.1.4.1

Move $x^{2}$ .

$\frac{8 (2 (x^{2} x^{2}) y^{4}) y^{3} - \sqrt[3]{8 x^{6} y^{12}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.4.2

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

$\frac{8 (2 x^{2 + 2} y^{4}) y^{3} - \sqrt[3]{8 x^{6} y^{12}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.4.3

Add $2$ and $2$ .

$\frac{8 (2 x^{4} y^{4}) y^{3} - \sqrt[3]{8 x^{6} y^{12}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.5

Multiply $y^{4}$ by $y^{3}$ by adding the exponents.

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Step 10.1.1.5.1

Move $y^{3}$ .

$\frac{8 (2 x^{4} (y^{3} y^{4})) - \sqrt[3]{8 x^{6} y^{12}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.5.2

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

$\frac{8 (2 x^{4} y^{3 + 4}) - \sqrt[3]{8 x^{6} y^{12}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.5.3

Add $3$ and $4$ .

$\frac{8 (2 x^{4} y^{7}) - \sqrt[3]{8 x^{6} y^{12}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.6

Multiply $2$ by $8$ .

$\frac{16 (x^{4} y^{7}) - \sqrt[3]{8 x^{6} y^{12}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.7

Rewrite $8 x^{6} y^{12}$ as ${(2 x^{2} y^{4})}^{3}$ .

$\frac{16 x^{4} y^{7} - \sqrt[3]{{(2 x^{2} y^{4})}^{3}} (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.8

Pull terms out from under the radical, assuming real numbers.

$\frac{16 x^{4} y^{7} - (2 x^{2} y^{4}) (8 y^{3} x^{2})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.9

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

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Step 10.1.1.9.1

Move $x^{2}$ .

$\frac{16 x^{4} y^{7} - (2 (x^{2} x^{2}) y^{4}) (8 y^{3})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.9.2

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

$\frac{16 x^{4} y^{7} - (2 x^{2 + 2} y^{4}) (8 y^{3})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.9.3

Add $2$ and $2$ .

$\frac{16 x^{4} y^{7} - (2 x^{4} y^{4}) (8 y^{3})}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.10

Multiply $y^{4}$ by $y^{3}$ by adding the exponents.

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Step 10.1.1.10.1

Move $y^{3}$ .

$\frac{16 x^{4} y^{7} - (2 x^{4} (y^{3} y^{4})) \cdot 8}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.10.2

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

$\frac{16 x^{4} y^{7} - (2 x^{4} y^{3 + 4}) \cdot 8}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.10.3

Add $3$ and $4$ .

$\frac{16 x^{4} y^{7} - (2 x^{4} y^{7}) \cdot 8}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.11

Multiply $2$ by $- 1$ .

$\frac{16 x^{4} y^{7} - 2 (x^{4} y^{7}) \cdot 8}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.1.12

Multiply $8$ by $- 2$ .

$\frac{16 x^{4} y^{7} - 16 x^{4} y^{7}}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.1.2

Subtract $16 x^{4} y^{7}$ from $16 x^{4} y^{7}$ .

$\frac{0}{{\sqrt{4 x^{4} y^{8}}}^{2}}$

Step 10.2

Combine terms.

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Step 10.2.1

Rewrite $\frac{0}{{\sqrt{4 x^{4} y^{8}}}^{2}}$ as $\frac{0}{{(2 x^{2} y^{4})}^{2}}$ .

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Step 10.2.1.1

Rewrite $4 x^{4} y^{8}$ as ${(2 x^{2} y^{4})}^{2}$ .

$\frac{0}{{\sqrt{{(2 x^{2} y^{4})}^{2}}}^{2}}$

Step 10.2.1.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{0}{{(2 x^{2} y^{4})}^{2}}$

Step 10.2.2

Factor $2$ out of $2 x^{2} y^{4}$ .

$\frac{0}{{(2 (x^{2} y^{4}))}^{2}}$

Step 10.2.3

Apply the product rule to $2 (x^{2} y^{4})$ .

$\frac{0}{2^{2} {(x^{2} y^{4})}^{2}}$

Step 10.2.4

Raise $2$ to the power of $2$ .

$\frac{0}{4 {(x^{2} y^{4})}^{2}}$

Step 10.2.5

Cancel the common factor of $0$ and $4$ .

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Step 10.2.5.1

Factor $4$ out of $0$ .

$\frac{4 (0)}{4 {(x^{2} y^{4})}^{2}}$

Step 10.2.5.2

Cancel the common factors.

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Step 10.2.5.2.1

Factor $4$ out of $4 {(x^{2} y^{4})}^{2}$ .

$\frac{4 (0)}{4 ({(x^{2} y^{4})}^{2})}$

Step 10.2.5.2.2

Cancel the common factor.

$\frac{4 \cdot 0}{4 {(x^{2} y^{4})}^{2}}$

Step 10.2.5.2.3

Rewrite the expression.

$\frac{0}{{(x^{2} y^{4})}^{2}}$

Step 10.3

Reorder terms.

$\frac{0}{{(y^{4} x^{2})}^{2}}$

Step 10.4

Simplify the denominator.

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Step 10.4.1

Apply the product rule to $y^{4} x^{2}$ .

$\frac{0}{{(y^{4})}^{2} {(x^{2})}^{2}}$

Step 10.4.2

Multiply the exponents in ${(y^{4})}^{2}$ .

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Step 10.4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{0}{y^{4 \cdot 2} {(x^{2})}^{2}}$

Step 10.4.2.2

Multiply $4$ by $2$ .

$\frac{0}{y^{8} {(x^{2})}^{2}}$

Step 10.4.3

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 10.4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{0}{y^{8} x^{2 \cdot 2}}$

Step 10.4.3.2

Multiply $2$ by $2$ .

$\frac{0}{y^{8} x^{4}}$

Step 10.5

Divide $0$ by $y^{8} x^{4}$ .

$0$