Simplify 3/(3-2 square root of 3)
The question is asking for the simplification of a given mathematical expression. Specifically, the expression is a rational expression (a fraction), where the numerator is the integer 3, and the denominator is the result of subtracting 2 times the square root of 3 from the integer 3. The goal is to manipulate the expression using algebraic techniques to reach a simpler or more standard form, typically one that no longer has a radical (square root) in the denominator.
$\frac{3}{3 - 2 \sqrt{3}}$
Answer
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:
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 $a - b$.
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.
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.
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 $3 + 2\sqrt{3}$.
Simplifying Expressions: This involves combining like terms and reducing expressions to their simplest form.
Negative Sign Distribution: When distributing a negative sign through a parenthesis, it changes the sign of each term inside the parenthesis.
