Problem 74 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ 2 \sqrt{\frac{6}{18}} $$

Step-by-Step Solution

Answer

The simplified radical expression is \(2 * \frac{\sqrt{3}}{3}\).

Step 1: Simplify The Fraction

Before anything else, simplify the fraction under the square root. Numerically, we reduce the fraction \(\frac{6}{18}\), which simplifies to \(\frac{1}{3}\). This makes the expression easier to work with.

Step 2: Insert Simplified Fraction back into Expression

Replace the original fraction with the simplified fraction in the expression. This changes the original expression to \(2 \sqrt{\frac{1}{3}} \).

Step 3: Use Properties of Radicals

Use the property of square roots that says the square root of a quotient is the quotient of the square roots. That is, \(\sqrt{\frac{1}{3}}\) simplifies to \(\frac{\sqrt{1}}{\sqrt{3}}\). So the expression now reads \(2 * \frac{\sqrt{1}}{\sqrt{3}}\).

Step 4: Simplify Radical

Simplify \(\sqrt{1}\) to 1 (because \(1^2\) equals to 1). This changes the expression to \(2 * \frac{1}{\sqrt{3}}\).

Step 5: Rationalize The Denominator

Multiply both the numerator and denominator by \(\sqrt{3}\) to rationalize the denominator. This results in \(2 * \frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}\), which simplifies to \(2 * \frac{\sqrt{3}}{3}\).

Key Concepts

Properties of Radicals

Understanding the properties of radicals is crucial when working with expressions involving roots. A radical, often represented as \( \sqrt[n]{x} \) for the n-th root of \( x \) (with \( \sqrt{x} \) being the square root when \( n = 2 \) ), has specific rules that help in simplifying expressions.

One vital property is the quotient property of radicals, which allows you to take the root of a fraction as the fraction of the roots. For example, \( \sqrt{\frac{a}{b}} \) simplifies to \( \frac{\sqrt{a}}{\sqrt{b}} \) when \( a \) and \( b \) are non-negative. This property is beneficial when simplifying square roots, as seen in steps 3 and 4 of the solution process.

• Quotient Property: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) when \( b eq 0 \) and \( a,b \geq 0 \)

Another important property related to simplifying is recognizing that the square root of a perfect square is an integer. For instance, \( \sqrt{1} \) is 1, as 1 is the square of 1.

Rationalize the Denominator

Rationalizing the denominator refers to the process of eliminating radicals from the denominator of a fraction. When a radical is present in the denominator, it can be difficult to work with, so transforming it into an integer or a rational number (hence 'rationalize') improves the expression's clarity.

The most common method to rationalize a square root in the denominator is to multiply both the numerator and the denominator by that same square root.

• Example: To rationalize \( \frac{1}{\sqrt{3}} \) , multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \) to get \( \frac{\sqrt{3}}{3} \) .

This doesn't change the value of the fraction, as \( \frac{\sqrt{3}}{\sqrt{3}} \) is 1, but it simplifies the form for further operations or for more intuitive understanding.

Simplify Square Roots

The procedure of simplifying square roots involves reducing the expression under the square root sign to its simplest form. This can mean factoring out perfect squares, separating the root of a fraction, or finding the square of an integer.

Remember that a square root 'undoes' a square. Therefore, \( \sqrt{x^2} \) equals \( x \) , assuming \( x \) is non-negative. When you encounter expressions like \( \sqrt{1} \) , it simplifies directly to 1, because 1 is \( 1^2 \) .

• Simplification: \( \sqrt{1} = 1 \) since \( 1^2 = 1 \)

Following the quotient property of radicals and knowing when you have a perfect square under the radical sign, as in step 4 of the solution, aids in the simplification process.

Algebraic Fractions

Algebraic fractions are simply fractions that contain variables. They follow the same principles as arithmetic fractions but are often involved in more complex manipulations due to the presence of variables.

The keys to working with algebraic fractions are simplifying and factoring expressions where possible, and understanding how to perform operations such as addition, subtraction, multiplication, and division.

• Simplify expressions by cancelling common factors in the numerator and denominator.
• When adding or subtracting, find a common denominator.
• Multiply by combining numerators and denominators, respectively.
• Divide by inverting the second fraction and multiplying.

Though the given exercise involves numerical values, the principles of handling algebraic fractions remain applicable, especially when generalizing techniques for expressions that include variables.