Algebra Examples

$p (x) = \frac{1}{\sqrt{x}}$ , $m (x) = x^{2} - 16$

Step 1

Set up the composite result function.

$m (p (x))$

Step 2

Evaluate $m (\frac{1}{\sqrt{x}})$ by substituting in the value of $p$ into $m$ .

$m (\frac{1}{\sqrt{x}}) = {(\frac{1}{\sqrt{x}})}^{2} - 16$

Step 3

Simplify each term.

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Step 3.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$m (\frac{1}{\sqrt{x}}) = {(\frac{1}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}})}^{2} - 16$

Step 3.2

Combine and simplify the denominator.

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Step 3.2.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$m (\frac{1}{\sqrt{x}}) = {(\frac{\sqrt{x}}{\sqrt{x} \sqrt{x}})}^{2} - 16$

Step 3.2.2

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

$m (\frac{1}{\sqrt{x}}) = {(\frac{\sqrt{x}}{\sqrt{x} \sqrt{x}})}^{2} - 16$

Step 3.2.3

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

$m (\frac{1}{\sqrt{x}}) = {(\frac{\sqrt{x}}{\sqrt{x} \sqrt{x}})}^{2} - 16$

Step 3.2.4

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

$m (\frac{1}{\sqrt{x}}) = {(\frac{\sqrt{x}}{{\sqrt{x}}^{1 + 1}})}^{2} - 16$

Step 3.2.5

Add $1$ and $1$ .

$m (\frac{1}{\sqrt{x}}) = {(\frac{\sqrt{x}}{{\sqrt{x}}^{2}})}^{2} - 16$

Step 3.2.6

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

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Step 3.2.6.1

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

$m (\frac{1}{\sqrt{x}}) = {(\frac{\sqrt{x}}{{(x^{\frac{1}{2}})}^{2}})}^{2} - 16$

Step 3.2.6.2

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

$m (\frac{1}{\sqrt{x}}) = {(\frac{\sqrt{x}}{x^{\frac{1}{2} \cdot 2}})}^{2} - 16$

Step 3.2.6.3

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

$m (\frac{1}{\sqrt{x}}) = {(\frac{\sqrt{x}}{x^{\frac{2}{2}}})}^{2} - 16$

Step 3.2.6.4

Cancel the common factor of $2$ .

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Step 3.2.6.4.1

Cancel the common factor.

$m (\frac{1}{\sqrt{x}}) = {(\frac{\sqrt{x}}{x^{\frac{2}{2}}})}^{2} - 16$

Step 3.2.6.4.2

Rewrite the expression.

$m (\frac{1}{\sqrt{x}}) = {(\frac{\sqrt{x}}{x})}^{2} - 16$

Step 3.2.6.5

Simplify.

$m (\frac{1}{\sqrt{x}}) = {(\frac{\sqrt{x}}{x})}^{2} - 16$

Step 3.3

Apply the product rule to $\frac{\sqrt{x}}{x}$ .

$m (\frac{1}{\sqrt{x}}) = \frac{{\sqrt{x}}^{2}}{x^{2}} - 16$

Step 3.4

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

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Step 3.4.1

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

$m (\frac{1}{\sqrt{x}}) = \frac{{(x^{\frac{1}{2}})}^{2}}{x^{2}} - 16$

Step 3.4.2

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

$m (\frac{1}{\sqrt{x}}) = \frac{x^{\frac{1}{2} \cdot 2}}{x^{2}} - 16$

Step 3.4.3

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

$m (\frac{1}{\sqrt{x}}) = \frac{x^{\frac{2}{2}}}{x^{2}} - 16$

Step 3.4.4

Cancel the common factor of $2$ .

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Step 3.4.4.1

Cancel the common factor.

$m (\frac{1}{\sqrt{x}}) = \frac{x^{\frac{2}{2}}}{x^{2}} - 16$

Step 3.4.4.2

Rewrite the expression.

$m (\frac{1}{\sqrt{x}}) = \frac{x}{x^{2}} - 16$

Step 3.4.5

Simplify.

$m (\frac{1}{\sqrt{x}}) = \frac{x}{x^{2}} - 16$

Step 3.5

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

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Step 3.5.1

Raise $x$ to the power of $1$ .

$m (\frac{1}{\sqrt{x}}) = \frac{x}{x^{2}} - 16$

Step 3.5.2

Factor $x$ out of $x^{1}$ .

$m (\frac{1}{\sqrt{x}}) = \frac{x \cdot 1}{x^{2}} - 16$

Step 3.5.3

Cancel the common factors.

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Step 3.5.3.1

Factor $x$ out of $x^{2}$ .

$m (\frac{1}{\sqrt{x}}) = \frac{x \cdot 1}{x \cdot x} - 16$

Step 3.5.3.2

Cancel the common factor.

$m (\frac{1}{\sqrt{x}}) = \frac{x \cdot 1}{x \cdot x} - 16$

Step 3.5.3.3

Rewrite the expression.

$m (\frac{1}{\sqrt{x}}) = \frac{1}{x} - 16$