simplify (sqrt(12))/(sqrt(3)+3)

1) Let's simplify:

\frac{\sqrt{12}}{\sqrt{3} + 3}

ⓘ We look for the largest square dividing

12

4

2^{2}

, and take it out of the root:

\sqrt{12}

\sqrt{4 \times 3}

\sqrt{4} \times \sqrt{3}

2 \times \sqrt{3}

\frac{2 \times \sqrt{3}}{\sqrt{3} + 3}

Multiply numerator and denominator by the conjugate

\sqrt{3} - 3

: (a+b)(a-b) = a²-b², here

{(\sqrt{3})}^{2} - 3^{2}

3 - 9

-6

ⅰ

\frac{2 \times \sqrt{3}}{\sqrt{3} + 3}

\frac{2 \times \sqrt{3} \times (\sqrt{3} - 3)}{-6}

\frac{2 \times \sqrt{3} \times (\sqrt{3} - 3)}{-6}

ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{2 \times \sqrt{3} \times (\sqrt{3} - 3)}{-6}

- \frac{2 \times \sqrt{3} \times (\sqrt{3} - 3)}{6}

- \frac{2 \times \sqrt{3} \times (\sqrt{3} - 3)}{6}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

- \frac{2 \times \sqrt{3} \times (\sqrt{3} - 3)}{6}

- \frac{\sqrt{3} \times (\sqrt{3} - 3)}{3}

- \frac{\sqrt{3} \times (\sqrt{3} - 3)}{3}

ⓘ Expand by multiplying each term inside the parentheses:

\sqrt{3} \times (\sqrt{3} - 3)

3 + \sqrt{3} \times (-3)

- \frac{3 + \sqrt{3} \times (-3)}{3}

ⓘ The minus sign is placed in front of the product:

\sqrt{3} \times (-3)

- 3 \times \sqrt{3}

- \frac{3 - 3 \times \sqrt{3}}{3}

ⓘ The minus sign of the fraction goes into the numerator or the denominator, changing the sign of each of its terms:

- \frac{3 - 3 \times \sqrt{3}}{3}

\frac{- 3 + 3 \times \sqrt{3}}{3}

\frac{- 3 + 3 \times \sqrt{3}}{3}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{- 3 + 3 \times \sqrt{3}}{3}

\frac{3 \times (- 1 + \sqrt{3})}{3}

- 1 + \sqrt{3}

- 1 + \sqrt{3}

▶Simplified form:

\sqrt{3} - 1

2) The numeric value of:

\sqrt{3} - 1

equals:

\approx 1.7321 - 1

▷

\approx 0.7321