Algebra Examples

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

Step 1

Rewrite $6 x^{10} y^{3}$ as ${(x^{5} y)}^{2} \cdot (6 y)$ .

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Step 1.1

Rewrite $x^{10}$ as ${(x^{5})}^{2}$ .

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

Step 1.2

Factor out $y^{2}$ .

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

Step 1.3

Move $6$ .

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

Step 1.4

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

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

Step 1.5

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $x^{5} y \sqrt{6 y}$ .

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

Step 3.2

Apply the product rule to $x^{5} y$ .

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

Step 4

Multiply the exponents in ${(x^{5})}^{5}$ .

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Step 4.1

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

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

Step 4.2

Multiply $5$ by $5$ .

$x^{25} y^{5} {\sqrt{6 y}}^{5}$

Step 5

Rewrite ${\sqrt{6 y}}^{5}$ as $\sqrt{{(6 y)}^{5}}$ .

$x^{25} y^{5} \sqrt{{(6 y)}^{5}}$

Step 6

Apply the product rule to $6 y$ .

$x^{25} y^{5} \sqrt{6^{5} y^{5}}$

Step 7

Raise $6$ to the power of $5$ .

$x^{25} y^{5} \sqrt{7776 y^{5}}$

Step 8

Rewrite $7776 y^{5}$ as ${(36 y^{2})}^{2} \cdot (6 y)$ .

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Step 8.1

Factor $1296$ out of $7776$ .

$x^{25} y^{5} \sqrt{1296 (6) y^{5}}$

Step 8.2

Rewrite $1296$ as $36^{2}$ .

$x^{25} y^{5} \sqrt{36^{2} \cdot 6 y^{5}}$

Step 8.3

Factor out $y^{4}$ .

$x^{25} y^{5} \sqrt{36^{2} \cdot 6 (y^{4} y)}$

Step 8.4

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

$x^{25} y^{5} \sqrt{36^{2} \cdot 6 ({(y^{2})}^{2} y)}$

Step 8.5

Move $6$ .

$x^{25} y^{5} \sqrt{36^{2} {(y^{2})}^{2} \cdot 6 y}$

Step 8.6

Rewrite $36^{2} {(y^{2})}^{2}$ as ${(36 y^{2})}^{2}$ .

$x^{25} y^{5} \sqrt{{(36 y^{2})}^{2} \cdot 6 y}$

Step 8.7

Add parentheses.

$x^{25} y^{5} \sqrt{{(36 y^{2})}^{2} \cdot (6 y)}$

Step 9

Pull terms out from under the radical.

$x^{25} y^{5} (36 y^{2} \sqrt{6 y})$

Step 10

Rewrite using the commutative property of multiplication.

$36 x^{25} y^{5} (y^{2} \sqrt{6 y})$

Step 11

Multiply $y^{5}$ by $y^{2}$ by adding the exponents.

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Step 11.1

Move $y^{2}$ .

$36 x^{25} (y^{2} y^{5}) \sqrt{6 y}$

Step 11.2

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

$36 x^{25} y^{2 + 5} \sqrt{6 y}$

Step 11.3

Add $2$ and $5$ .

$36 x^{25} y^{7} \sqrt{6 y}$