Simplify ( square root of x-2+3(x-2)^(1/2))/(x^2-4x+4)

$\frac{\sqrt{x - 2} + 3 {(x - 2)}^{\frac{1}{2}}}{x^{2} - 4 x + 4}$

Simplify the numerator.

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

$\frac{{(x - 2)}^{\frac{1}{2}} + 3 {(x - 2)}^{\frac{1}{2}}}{x^{2} - 4 x + 4}$

Add ${(x - 2)}^{\frac{1}{2}}$ and $3 {(x - 2)}^{\frac{1}{2}}$ .

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

Factor using the perfect square rule.

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

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Move ${(x - 2)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Multiply ${(x - 2)}^{2}$ by ${(x - 2)}^{- \frac{1}{2}}$ by adding the exponents.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

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

Subtract $1$ from $4$ .

$\frac{4}{{(x - 2)}^{\frac{3}{2}}}$