Calculus Examples

${(\sqrt{x^{9} y})}^{6}$

Step 1

Rewrite $x^{9} y$ as ${({(x^{2})}^{2})}^{2} (x y)$ .

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Step 1.1

Factor out $x^{8}$ .

${\sqrt{x^{8} x y}}^{6}$

Step 1.2

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

${\sqrt{{(x^{4})}^{2} x y}}^{6}$

Step 1.3

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

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

Step 1.4

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Multiply $2$ by $2$ .

${(x^{4} \sqrt{x y})}^{6}$

Step 3.2

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

${(x^{4})}^{6} {\sqrt{x y}}^{6}$

Step 3.3

Multiply the exponents in ${(x^{4})}^{6}$ .

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Step 3.3.1

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

$x^{4 \cdot 6} {\sqrt{x y}}^{6}$

Step 3.3.2

Multiply $4$ by $6$ .

$x^{24} {\sqrt{x y}}^{6}$

Step 4

Rewrite ${\sqrt{x y}}^{6}$ as ${(x y)}^{3}$ .

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x y}$ as ${(x y)}^{\frac{1}{2}}$ .

$x^{24} {({(x y)}^{\frac{1}{2}})}^{6}$

Step 4.2

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

$x^{24} {(x y)}^{\frac{1}{2} \cdot 6}$

Step 4.3

Combine $\frac{1}{2}$ and $6$ .

$x^{24} {(x y)}^{\frac{6}{2}}$

Step 4.4

Cancel the common factor of $6$ and $2$ .

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Step 4.4.1

Factor $2$ out of $6$ .

$x^{24} {(x y)}^{\frac{2 \cdot 3}{2}}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $2$ .

$x^{24} {(x y)}^{\frac{2 \cdot 3}{2 (1)}}$

Step 4.4.2.2

Cancel the common factor.

$x^{24} {(x y)}^{\frac{2 \cdot 3}{2 \cdot 1}}$

Step 4.4.2.3

Rewrite the expression.

$x^{24} {(x y)}^{\frac{3}{1}}$

Step 4.4.2.4

Divide $3$ by $1$ .

$x^{24} {(x y)}^{3}$

Step 5

Apply the product rule to $x y$ .

$x^{24} (x^{3} y^{3})$

Step 6

Multiply $x^{24}$ by $x^{3}$ by adding the exponents.

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Step 6.1

Move $x^{3}$ .

$x^{3} x^{24} y^{3}$

Step 6.2

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

$x^{3 + 24} y^{3}$

Step 6.3

Add $3$ and $24$ .

$x^{27} y^{3}$