Simplify ( square root of 3-2)/(5-2 square root of 3)

$\frac{\sqrt{3} - 2}{5 - 2 \sqrt{3}}$

Step 1

Multiply $\frac{\sqrt{3} - 2}{5 - 2 \sqrt{3}}$ by $\frac{5 + 2 \sqrt{3}}{5 + 2 \sqrt{3}}$ .

$\frac{\sqrt{3} - 2}{5 - 2 \sqrt{3}} \cdot \frac{5 + 2 \sqrt{3}}{5 + 2 \sqrt{3}}$

Step 2

Combine fractions.

Multiply $\frac{\sqrt{3} - 2}{5 - 2 \sqrt{3}}$ by $\frac{5 + 2 \sqrt{3}}{5 + 2 \sqrt{3}}$ .

$\frac{(\sqrt{3} - 2) (5 + 2 \sqrt{3})}{(5 - 2 \sqrt{3}) (5 + 2 \sqrt{3})}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{3} - 2) (5 + 2 \sqrt{3})}{25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 {\sqrt{3}}^{2}}$

Simplify.

$\frac{(\sqrt{3} - 2) (5 + 2 \sqrt{3})}{13}$

Step 3

Expand $(\sqrt{3} - 2) (5 + 2 \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{3} (5 + 2 \sqrt{3}) - 2 (5 + 2 \sqrt{3})}{13}$

Apply the distributive property.

$\frac{\sqrt{3} \cdot 5 + \sqrt{3} (2 \sqrt{3}) - 2 (5 + 2 \sqrt{3})}{13}$

Apply the distributive property.

$\frac{\sqrt{3} \cdot 5 + \sqrt{3} (2 \sqrt{3}) - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

Step 4

Simplify and combine like terms.

Simplify each term.

Move $5$ to the left of $\sqrt{3}$ .

$\frac{5 \cdot \sqrt{3} + \sqrt{3} (2 \sqrt{3}) - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

Multiply $\sqrt{3} (2 \sqrt{3})$ .

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{5 \sqrt{3} + 2 ({\sqrt{3}}^{1} \sqrt{3}) - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{5 \sqrt{3} + 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

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

$\frac{5 \sqrt{3} + 2 {\sqrt{3}}^{1 + 1} - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

Add $1$ and $1$ .

$\frac{5 \sqrt{3} + 2 {\sqrt{3}}^{2} - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{5 \sqrt{3} + 2 {(3^{\frac{1}{2}})}^{2} - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

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

$\frac{5 \sqrt{3} + 2 \cdot 3^{\frac{1}{2} \cdot 2} - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

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

$\frac{5 \sqrt{3} + 2 \cdot 3^{\frac{2}{2}} - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 \sqrt{3} + 2 \cdot 3^{\frac{2}{2}} - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

Rewrite the expression.

$\frac{5 \sqrt{3} + 2 \cdot 3^{1} - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

Evaluate the exponent.

$\frac{5 \sqrt{3} + 2 \cdot 3 - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

Multiply $2$ by $3$ .

$\frac{5 \sqrt{3} + 6 - 2 \cdot 5 - 2 (2 \sqrt{3})}{13}$

Multiply $- 2$ by $5$ .

$\frac{5 \sqrt{3} + 6 - 10 - 2 (2 \sqrt{3})}{13}$

Multiply $2$ by $- 2$ .

$\frac{5 \sqrt{3} + 6 - 10 - 4 \sqrt{3}}{13}$

Subtract $4 \sqrt{3}$ from $5 \sqrt{3}$ .

$\frac{6 - 10 + \sqrt{3}}{13}$

Subtract $10$ from $6$ .

$\frac{- 4 + \sqrt{3}}{13}$

Step 5

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- 1 (4) + \sqrt{3}}{13}$

Step 6

Factor $- 1$ out of $\sqrt{3}$ .

$\frac{- 1 (4) - 1 (- \sqrt{3})}{13}$

Step 7

Factor $- 1$ out of $- 1 (4) - 1 (- \sqrt{3})$ .

$\frac{- 1 (4 - \sqrt{3})}{13}$

Step 8

Move the negative in front of the fraction.

$- \frac{4 - \sqrt{3}}{13}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 - \sqrt{3}}{13}$

Decimal Form:

$- 0.17445763 \dots$

Simplify ( square root of 3-2)/(5-2 square root of 3)

Related Questions

Support LCMGCF.com by disabling your adblocker.