Calculus Examples

$y = \sqrt{\frac{x}{2} + 16}$

Step 1

Simplify $\sqrt{\frac{x}{2} + 16}$ .

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Step 1.1

To write $16$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \sqrt{\frac{x}{2} + 16 \cdot \frac{2}{2}}$

Step 1.2

Combine $16$ and $\frac{2}{2}$ .

$y = \sqrt{\frac{x}{2} + \frac{16 \cdot 2}{2}}$

Step 1.3

Combine the numerators over the common denominator.

$y = \sqrt{\frac{x + 16 \cdot 2}{2}}$

Step 1.4

Multiply $16$ by $2$ .

$y = \sqrt{\frac{x + 32}{2}}$

Step 1.5

Rewrite $\sqrt{\frac{x + 32}{2}}$ as $\frac{\sqrt{x + 32}}{\sqrt{2}}$ .

$y = \frac{\sqrt{x + 32}}{\sqrt{2}}$

Step 1.6

Multiply $\frac{\sqrt{x + 32}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \frac{\sqrt{x + 32}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

Multiply $\frac{\sqrt{x + 32}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \frac{\sqrt{x + 32} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$y = \frac{\sqrt{x + 32} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$y = \frac{\sqrt{x + 32} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.7.4

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

$y = \frac{\sqrt{x + 32} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.7.5

Add $1$ and $1$ .

$y = \frac{\sqrt{x + 32} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.7.6

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

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Step 1.7.6.1

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

$y = \frac{\sqrt{x + 32} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.7.6.2

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

$y = \frac{\sqrt{x + 32} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.7.6.3

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

$y = \frac{\sqrt{x + 32} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.7.6.4

Cancel the common factor of $2$ .

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Step 1.7.6.4.1

Cancel the common factor.

$y = \frac{\sqrt{x + 32} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.7.6.4.2

Rewrite the expression.

$y = \frac{\sqrt{x + 32} \sqrt{2}}{2^{1}}$

Step 1.7.6.5

Evaluate the exponent.

$y = \frac{\sqrt{x + 32} \sqrt{2}}{2}$

Step 1.8

Combine using the product rule for radicals.

$y = \frac{\sqrt{(x + 32) \cdot 2}}{2}$

Step 1.9

Reorder factors in $\frac{\sqrt{(x + 32) \cdot 2}}{2}$ .

$y = \frac{\sqrt{2 (x + 32)}}{2}$

Step 2

To rewrite as a function of $x$ , write the equation so that $y$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = \frac{\sqrt{2 (x + 32)}}{2}$