Question

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{2}{\sqrt{3}}$$

Short answer

\frac{2\frac{3}}{3} = \frac{2\frac{3}}{3} -> \frac{2}{3}

Step-by-step solution

Understand the Problem

The goal is to eliminate the square root in the denominator of the fraction \(\frac{2}{\frac3}\). This process is called rationalizing the denominator.

Multiply the Numerator and Denominator by the Same Radical

To rationalize \(\frac{2}{\frac{3}}\), multiply both the numerator and the denominator by the square root that appears in the denominator. That means multiplying by \(\frac{3}}\): \(\frac{2}{\frac{3}} \times \frac{\frac{3}}{\frac{3}}\)

Simplify the Fraction

Perform the multiplication: \(\frac{2 \times \frac{3}}{\frac{3} \times \frac{3}} = \frac{2\frac{3}}{3}\)

Final Simplification

Simplify the expression \(\frac{2\frac{3}}{3}\) to \(\frac{2}{3}\). Now the fraction \(\frac{2}{\frac{3}}\) is rationalized.

Key concepts

Square Roots

Square roots are a fundamental concept in mathematics. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\).
The symbol for square root is \(\backslashmathrm{sqrt}\(\backslashcdot\)\).
If you have \(\backslashsqrt{a}\), it means you're looking for a number which, when squared, equals \a\. Square roots are important for simplifying expressions, especially when working with exponents and equations.

Key points to remember about square roots:

• The square root of a positive number is always positive.
• Every positive number has two square roots: \pm \backslashsqrt{a} \, but in most practical scenarios, we consider only the principal (positive) square root.
• Square roots of perfect squares (like 1, 4, 9, 16) are whole numbers.

Fractions

Fractions are expressions that represent parts of a whole. They are composed of a numerator (top part) and a denominator (bottom part).
For example, in the fraction \(\backslashfrac{2}{3}\), the number 2 is the numerator and 3 is the denominator.

Here are some important aspects of fractions:

• Simplifying fractions: To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).
• Rationalizing the denominator: Sometimes, fractions have irrational numbers (like square roots) in their denominators. To rationalize them, we multiply both the numerator and the denominator by a suitable radical that can eliminate the square root in the denominator.

Using fractions is crucial when dealing with various calculations in mathematics. This not only makes calculations easier but also keeps the expressions clear and concise. In our example, we are seeking to rationalize \(\backslashfrac{2}{\backslashsqrt{3}\}\).

Simplification

Simplification is the process of making an expression as simple as possible. In math, an expression is simplified by performing operations and combining like terms to make it easier to understand and use.

This includes:

• Combining like terms: Terms that have the same variables raised to the same power can be combined. For example, \ 2x + 3x = 5x\.
  
• Performing calculations: This involves carrying out arithmetic operations whenever possible, like \ 2 + 3 = 5\.
  
• Rationalizing denominators: This specific type of simplification eliminates radicals from the denominator, which often makes the fractions easier to work with.

For our exercise, simplifying \(\backslashfrac{2 \backslashsqrt{3}}{\backslashsqrt{3} \backslashsqrt{3}}\) step by step finally allows us to write the simplified form as \(\backslashfrac{2 \backslashsqrt{3}}{3}\). Each step has a purpose, clearing unnecessary complexity bit by bit.