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

Solution:

Step:1

Rationalize the denominator of $\frac{3}{3-2\sqrt{3}}$ by multiplying by the conjugate $\frac{3+2\sqrt{3}}{3+2\sqrt{3}}$.

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

Step:2

Apply the multiplication to the numerators.

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

Step:3

Use the difference of squares to expand the denominator.

$$\frac{3\cdot (3+2\sqrt{3})}{9-(2\sqrt{3}{)}^2}$$

Step:4

Simplify the denominator.

$$\frac{3\cdot (3+2\sqrt{3})}{9-12}$$

Step:5

Further simplify the expression.

Step:5.1

Isolate the negative sign from the denominator.

$$-1\cdot (3+2\sqrt{3})$$

Step:5.2

Express the multiplication by the negative sign.

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

Step:6

Distribute the negative sign.

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

Step:7

Final multiplication.

Step:7.1

Multiply $-1$ by $3$.

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

Step:7.2

Multiply $2$ by $\sqrt{3}$ and apply the negative sign.

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

Step:8

Present the final result in various forms.

Exact Form: $-3-2\sqrt{3}$ Decimal Form: Approximately $-6.46410161$

Knowledge Notes:

1. Rationalizing the Denominator: This process involves multiplying the numerator and the denominator by the conjugate of the denominator to eliminate the square root from the denominator. The conjugate of a binomial $a+b$ is $a-b$.

2. Difference of Squares: This is a pattern used in algebra where $(a+b)(a-b)=a^2-b^2$. It is used to simplify expressions where two terms are being multiplied by their conjugates.

3. Distributive Property: This property states that $a(b+c)=ab+ac$. It is used to multiply a single term by each term inside a parenthesis.

4. Conjugate: The conjugate of a binomial is obtained by changing the sign between two terms. For example, the conjugate of $3-2\sqrt{3}$ is $3+2\sqrt{3}$.

5. Simplifying Expressions: This involves combining like terms and reducing expressions to their simplest form.

6. Negative Sign Distribution: When distributing a negative sign through a parenthesis, it changes the sign of each term inside the parenthesis.