How to Simplify Radical Expressions Involving Fractions

1) Simplify: $\frac{\sqrt{125 \ n^5}}{\sqrt{5 \ n}}$

$\color{red}{\frac{\sqrt{125 \ n^5}}{\sqrt{5 \ n}} \ = \ \frac{\sqrt{5^2 \times 5 \ n^{2 \times 2} \times n}}{\sqrt{5 \ n}} \ = \ \sqrt{5^2 \ n^{2 \times 2}} \ = \ 5 \ n^2}$

2) Simplify: $\frac{\sqrt{108 \ x^{11}}}{\sqrt{6 \ x^2}}$

$\color{red}{\frac{\sqrt{108 \ x^{11}}}{\sqrt{6 \ x^2}} \ = \ \frac{\sqrt{3^3 \times 2^2 \ x^{2 \times 4} \times x^2 \times x}}{\sqrt{2 \times 3 \ x^2}} \ = \ \sqrt{3^2 \ x^{2 \times 4} \times x} \ = \ 3 \ x^4 \sqrt{x}}$

3) Simplify: $\frac{\sqrt{66 \ n^6}}{\sqrt{22 \ n^2}}$

$\color{red}{\frac{\sqrt{66 \ n^6}}{\sqrt{22 \ n^2}} \ = \ \frac{\sqrt{6 \times 11 \ n^{2 \times 3}}}{\sqrt{2 \times 11 \ n^{2 \times 1}}} \ = \ \sqrt{6 \ n^{2 \times 2}} \ = \ n \ \sqrt{6}}$

4) Simplify: $\frac{5}{\sqrt{3}}$

$\color{red}{\frac{5}{\sqrt{3}} \ = \ \frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \ = \ \frac{5 \ \sqrt{3}}{3}}$

5) Simplify: $\frac{6 \ \sqrt{2x}}{\sqrt{x}}$

$\color{red}{\frac{6 \ \sqrt{2x}}{\sqrt{x}} \ = \ \frac{6 \ \sqrt{2x}}{\sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} \ = \ \frac{6 \ \sqrt{2 \ x^2}}{x} \ = \ \frac{6 \ x \ \sqrt{2}}{x} \ = \ 6 \ \sqrt{2}}$

6) Simplify: $\frac{3 \ \sqrt{72 \ x^3}}{\sqrt{12x}}$

$\color{red}{\frac{3 \ \sqrt{72 \ x^3}}{\sqrt{12x}} \ = \ 3 \ \sqrt{\frac{72 \ x^3}{12x}} \ = \ 3 \ \sqrt{6 \ x^2} \ = \ 3 \ x \ \sqrt{6}}$

7) Simplify: $\frac{\sqrt{70 \ x^7}}{\sqrt{5x^2}}$

$\color{red}{\frac{\sqrt{70 \ x^7}}{\sqrt{5x^2}} \ = \ \sqrt{\frac{70 \ x^7}{5x^2}} \ = \ \sqrt{14 \ x^5} \ = \ x^2 \ \sqrt{14x}}$

8) Simplify: $\frac{\sqrt{36 \ x^2}}{\sqrt{12x^5}}$

$\color{red}{\frac{\sqrt{36 \ x^2}}{\sqrt{12x^5}} \ = \ \sqrt{\frac{36 \ x^2}{12x^5}} \ = \ \sqrt{\frac{3}{x^3}} \ = \ \frac{\sqrt{3}}{x \ \sqrt{x}} \ = \ \frac{\sqrt{3}}{x \ \sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} \ = \ \frac{\sqrt{3x}}{x \ \sqrt{x^2}} \ = \ \frac{\sqrt{3x}}{x^2}}$

9) Simplify: $\frac{\sqrt{69 \ m^7}}{\sqrt{23 \ m^{12}}}$

$\color{red}{\frac{\sqrt{69 \ m^7}}{\sqrt{23 \ m^{12}}} \ = \ \sqrt{\frac{69 \ m^7}{23 \ m^{12}}} \ = \ \sqrt{\frac{3}{m^5}} \ = \ \frac{\sqrt{3}}{m^2 \ \sqrt{m}} \ = \ \frac{\sqrt{3}}{m^2 \ \sqrt{m}} \times \frac{\sqrt{m}}{\sqrt{m}} \ = \ \frac{\sqrt{3m}}{m^2 \ \sqrt{m^2}} \ = \ \frac{\sqrt{3m}}{m^3}}$

10) Simplify: $\frac{7}{1 \ + \ \sqrt{z}}$

$\color{red}{\frac{7}{1 \ + \ \sqrt{z}} \ = \ \frac{7}{1 \ + \ \sqrt{z}} \times \frac{1 \ - \ \sqrt{z}}{1 \ - \ \sqrt{z}} \ = \ \frac{7 \ - \ 7 \ \sqrt{z}}{1 \ - \ z}}$