Finite Math Examples

${(\sqrt{4 x^{2} y^{3}})}^{4}$

Step 1

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

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Step 1.1

Rewrite $4$ as $2^{2}$ .

${\sqrt{2^{2} x^{2} y^{3}}}^{4}$

Step 1.2

Factor out $y^{2}$ .

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

Step 1.3

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

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

Step 2

Pull terms out from under the radical.

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

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 $2 x y \sqrt{y}$ .

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

Step 3.2

Apply the product rule to $2 x y$ .

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

Step 3.3

Apply the product rule to $2 x$ .

$2^{4} x^{4} y^{4} {\sqrt{y}}^{4}$

Step 4

Raise $2$ to the power of $4$ .

$16 x^{4} y^{4} {\sqrt{y}}^{4}$

Step 5

Rewrite ${\sqrt{y}}^{4}$ as $y^{2}$ .

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Step 5.1

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

$16 x^{4} y^{4} {(y^{\frac{1}{2}})}^{4}$

Step 5.2

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

$16 x^{4} y^{4} y^{\frac{1}{2} \cdot 4}$

Step 5.3

Combine $\frac{1}{2}$ and $4$ .

$16 x^{4} y^{4} y^{\frac{4}{2}}$

Step 5.4

Cancel the common factor of $4$ and $2$ .

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Step 5.4.1

Factor $2$ out of $4$ .

$16 x^{4} y^{4} y^{\frac{2 \cdot 2}{2}}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $2$ out of $2$ .

$16 x^{4} y^{4} y^{\frac{2 \cdot 2}{2 (1)}}$

Step 5.4.2.2

Cancel the common factor.

$16 x^{4} y^{4} y^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 5.4.2.3

Rewrite the expression.

$16 x^{4} y^{4} y^{\frac{2}{1}}$

Step 5.4.2.4

Divide $2$ by $1$ .

$16 x^{4} y^{4} y^{2}$

Step 6

Multiply $y^{4}$ by $y^{2}$ by adding the exponents.

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Step 6.1

Move $y^{2}$ .

$16 x^{4} (y^{2} y^{4})$

Step 6.2

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

$16 x^{4} y^{2 + 4}$

Step 6.3

Add $2$ and $4$ .

$16 x^{4} y^{6}$