Calculus Examples

${(\frac{16 x^{2} y^{2}}{3 x^{- 2} y^{4}})}^{\frac{1}{2}}$

Step 1

Move $x^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{16 x^{2} y^{2} x^{2}}{3 y^{4}})}^{\frac{1}{2}}$

Step 2

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

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Step 2.1

Move $x^{2}$ .

${(\frac{16 (x^{2} x^{2}) y^{2}}{3 y^{4}})}^{\frac{1}{2}}$

Step 2.2

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

${(\frac{16 x^{2 + 2} y^{2}}{3 y^{4}})}^{\frac{1}{2}}$

Step 2.3

Add $2$ and $2$ .

${(\frac{16 x^{4} y^{2}}{3 y^{4}})}^{\frac{1}{2}}$

Step 3

Cancel the common factor of $y^{2}$ and $y^{4}$ .

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Step 3.1

Factor $y^{2}$ out of $16 x^{4} y^{2}$ .

${(\frac{y^{2} (16 x^{4})}{3 y^{4}})}^{\frac{1}{2}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $y^{2}$ out of $3 y^{4}$ .

${(\frac{y^{2} (16 x^{4})}{y^{2} (3 y^{2})})}^{\frac{1}{2}}$

Step 3.2.2

Cancel the common factor.

${(\frac{y^{2} (16 x^{4})}{y^{2} (3 y^{2})})}^{\frac{1}{2}}$

Step 3.2.3

Rewrite the expression.

${(\frac{16 x^{4}}{3 y^{2}})}^{\frac{1}{2}}$

Step 4

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

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Step 4.1

Apply the product rule to $\frac{16 x^{4}}{3 y^{2}}$ .

$\frac{{(16 x^{4})}^{\frac{1}{2}}}{{(3 y^{2})}^{\frac{1}{2}}}$

Step 4.2

Apply the product rule to $16 x^{4}$ .

$\frac{16^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}{{(3 y^{2})}^{\frac{1}{2}}}$

Step 4.3

Apply the product rule to $3 y^{2}$ .

$\frac{16^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}{3^{\frac{1}{2}} {(y^{2})}^{\frac{1}{2}}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $16$ as $4^{2}$ .

$\frac{{(4^{2})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}{3^{\frac{1}{2}} {(y^{2})}^{\frac{1}{2}}}$

Step 5.2

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

$\frac{4^{2 (\frac{1}{2})} {(x^{4})}^{\frac{1}{2}}}{3^{\frac{1}{2}} {(y^{2})}^{\frac{1}{2}}}$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Cancel the common factor.

$\frac{4^{2 (\frac{1}{2})} {(x^{4})}^{\frac{1}{2}}}{3^{\frac{1}{2}} {(y^{2})}^{\frac{1}{2}}}$

Step 5.3.2

Rewrite the expression.

$\frac{4^{1} {(x^{4})}^{\frac{1}{2}}}{3^{\frac{1}{2}} {(y^{2})}^{\frac{1}{2}}}$

Step 5.4

Evaluate the exponent.

$\frac{4 {(x^{4})}^{\frac{1}{2}}}{3^{\frac{1}{2}} {(y^{2})}^{\frac{1}{2}}}$

Step 5.5

Multiply the exponents in ${(x^{4})}^{\frac{1}{2}}$ .

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Step 5.5.1

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

$\frac{4 x^{4 (\frac{1}{2})}}{3^{\frac{1}{2}} {(y^{2})}^{\frac{1}{2}}}$

Step 5.5.2

Cancel the common factor of $2$ .

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Step 5.5.2.1

Factor $2$ out of $4$ .

$\frac{4 x^{2 (2) \frac{1}{2}}}{3^{\frac{1}{2}} {(y^{2})}^{\frac{1}{2}}}$

Step 5.5.2.2

Cancel the common factor.

$\frac{4 x^{2 \cdot 2 \frac{1}{2}}}{3^{\frac{1}{2}} {(y^{2})}^{\frac{1}{2}}}$

Step 5.5.2.3

Rewrite the expression.

$\frac{4 x^{2}}{3^{\frac{1}{2}} {(y^{2})}^{\frac{1}{2}}}$

Step 6

Simplify the denominator.

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Step 6.1

Multiply the exponents in ${(y^{2})}^{\frac{1}{2}}$ .

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Step 6.1.1

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

$\frac{4 x^{2}}{3^{\frac{1}{2}} y^{2 (\frac{1}{2})}}$

Step 6.1.2

Cancel the common factor of $2$ .

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Step 6.1.2.1

Cancel the common factor.

$\frac{4 x^{2}}{3^{\frac{1}{2}} y^{2 (\frac{1}{2})}}$

Step 6.1.2.2

Rewrite the expression.

$\frac{4 x^{2}}{3^{\frac{1}{2}} y^{1}}$

Step 6.2

Simplify.

$\frac{4 x^{2}}{3^{\frac{1}{2}} y}$

Step 7

Reorder factors in $\frac{4 x^{2}}{3^{\frac{1}{2}} y}$ .

$\frac{4 x^{2}}{y \cdot 3^{\frac{1}{2}}}$