Calculus Examples

${(\frac{1}{2} x^{- \frac{1}{2}})}^{2}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Combine.

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

Step 3

Multiply $1$ by $1$ .

${(\frac{1}{2 x^{\frac{1}{2}}})}^{2}$

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 $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{1^{2}}{{(2 x^{\frac{1}{2}})}^{2}}$

Step 4.2

Apply the product rule to $2 x^{\frac{1}{2}}$ .

$\frac{1^{2}}{2^{2} {(x^{\frac{1}{2}})}^{2}}$

Step 5

One to any power is one.

$\frac{1}{2^{2} {(x^{\frac{1}{2}})}^{2}}$

Step 6

Simplify the denominator.

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factor of $2$ .

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Step 6.2.2.1

Cancel the common factor.

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

Step 6.2.2.2

Rewrite the expression.

$\frac{1}{4 x^{1}}$

Step 6.3

Simplify.

$\frac{1}{4 x}$