Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

$x^{2} {(x^{2} + 2)}^{1}$

Step 3

Simplify.

$x^{2} (x^{2} + 2)$

Step 4

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 2$

Step 5

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

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Step 5.1

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

$x^{2 + 2} + x^{2} \cdot 2$

Step 5.2

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot 2$

Step 6

Move $2$ to the left of $x^{2}$ .

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