Find p(m(x)) p(x)=1/( square root of x) , m(x)=x^2-16

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

Step 1

Set up the composite result function.

$p (m (x))$

Step 2

Evaluate $p (x^{2} - 16)$ by substituting in the value of $m$ into $p$ .

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

Step 3

Remove parentheses.

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

Step 4

Simplify the denominator.

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

$p (x^{2} - 16) = \frac{1}{\sqrt{x^{2} - 4^{2}}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$p (x^{2} - 16) = \frac{1}{\sqrt{(x + 4) (x - 4)}}$

Step 5

Multiply $\frac{1}{\sqrt{(x + 4) (x - 4)}}$ by $\frac{\sqrt{(x + 4) (x - 4)}}{\sqrt{(x + 4) (x - 4)}}$ .

$p (x^{2} - 16) = \frac{1}{\sqrt{(x + 4) (x - 4)}} \cdot \frac{\sqrt{(x + 4) (x - 4)}}{\sqrt{(x + 4) (x - 4)}}$

Step 6

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{(x + 4) (x - 4)}}$ by $\frac{\sqrt{(x + 4) (x - 4)}}{\sqrt{(x + 4) (x - 4)}}$ .

$p (x^{2} - 16) = \frac{\sqrt{(x + 4) (x - 4)}}{\sqrt{(x + 4) (x - 4)} \sqrt{(x + 4) (x - 4)}}$

Raise $\sqrt{(x + 4) (x - 4)}$ to the power of $1$ .

$p (x^{2} - 16) = \frac{\sqrt{(x + 4) (x - 4)}}{\sqrt{(x + 4) (x - 4)} \sqrt{(x + 4) (x - 4)}}$

Raise $\sqrt{(x + 4) (x - 4)}$ to the power of $1$ .

$p (x^{2} - 16) = \frac{\sqrt{(x + 4) (x - 4)}}{\sqrt{(x + 4) (x - 4)} \sqrt{(x + 4) (x - 4)}}$

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

$p (x^{2} - 16) = \frac{\sqrt{(x + 4) (x - 4)}}{{\sqrt{(x + 4) (x - 4)}}^{1 + 1}}$

Add $1$ and $1$ .

$p (x^{2} - 16) = \frac{\sqrt{(x + 4) (x - 4)}}{{\sqrt{(x + 4) (x - 4)}}^{2}}$

Rewrite ${\sqrt{(x + 4) (x - 4)}}^{2}$ as $(x + 4) (x - 4)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 4) (x - 4)}$ as ${((x + 4) (x - 4))}^{\frac{1}{2}}$ .

$p (x^{2} - 16) = \frac{\sqrt{(x + 4) (x - 4)}}{{({((x + 4) (x - 4))}^{\frac{1}{2}})}^{2}}$

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

$p (x^{2} - 16) = \frac{\sqrt{(x + 4) (x - 4)}}{{((x + 4) (x - 4))}^{\frac{1}{2} \cdot 2}}$

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

$p (x^{2} - 16) = \frac{\sqrt{(x + 4) (x - 4)}}{{((x + 4) (x - 4))}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$p (x^{2} - 16) = \frac{\sqrt{(x + 4) (x - 4)}}{{((x + 4) (x - 4))}^{\frac{2}{2}}}$

Rewrite the expression.

$p (x^{2} - 16) = \frac{\sqrt{(x + 4) (x - 4)}}{(x + 4) (x - 4)}$

Simplify.

$p (x^{2} - 16) = \frac{\sqrt{(x + 4) (x - 4)}}{(x + 4) (x - 4)}$