Solve (x+1)(1/2x^-frac{1{2}})+(x^frac{1{2}}+3)(1)

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

Evaluate

$$\frac{3\sqrt{x}}{2}+3+\frac{1}{2\sqrt{x}}$$

Solution Steps

Use the distributive property to multiply x+1 by \frac{1}{2}.

$$\left(\frac{1}{2}x+\frac{1}{2}\right)x^{-\frac{1}{2}}+\left(x^{\frac{1}{2}}+3\right)\times 1$$

Use the distributive property to multiply \frac{1}{2}x+\frac{1}{2} by x^{-\frac{1}{2}}.

$$\frac{1}{2}xx^{-\frac{1}{2}}+\frac{1}{2}x^{-\frac{1}{2}}+\left(x^{\frac{1}{2}}+3\right)\times 1$$

To multiply powers of the same base, add their exponents. Add 1 and -\frac{1}{2} to get \frac{1}{2}.

$$\frac{1}{2}x^{\frac{1}{2}}+\frac{1}{2}x^{-\frac{1}{2}}+\left(x^{\frac{1}{2}}+3\right)\times 1$$

Use the distributive property to multiply x^{\frac{1}{2}}+3 by 1.

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

Combine \frac{1}{2}x^{\frac{1}{2}} and x^{\frac{1}{2}} to get \frac{3}{2}x^{\frac{1}{2}}.

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

Factor

$$\frac{3\sqrt{x}+6+\frac{1}{\sqrt{x}}}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{\frac{x+1}{\sqrt{x}}+2\sqrt{x}+6}{2}$$