Problem 57 - Intermediate Algebra

Question

Rationalize each denominator. $$ \sqrt{\frac{8}{3}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{2\sqrt{6}}{3} \).

Step 1: Identify the Radical in the Denominator

We are given the expression \( \sqrt{\frac{8}{3}} \). To rationalize the denominator, we need to eliminate the square root in the denominator. Currently, the expression does not display the denominator clearly, but under typical evaluation, it implies a multiplying effect by dividing \( \sqrt{8} \) by \( \sqrt{3} \).

Step 2: Rationalize the Denominator

To remove the square root in the denominator, multiply both the numerator and the denominator by \( \sqrt{3} \). This will give us \( \sqrt{\frac{8}{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{24}}{3} \).

Step 3: Simplify the Expression

Simplify \( \sqrt{24} \) in the numerator. Notice that \( 24 = 4 \times 6 = 2^2 \times 6 \), so \( \sqrt{24} = \sqrt{2^2 \times 6} = 2\sqrt{6} \). Thus, the expression becomes \( \frac{2\sqrt{6}}{3} \).

Step 4: Finalize the Simplified Expression

Now, write the final simplified expression without a radical in the denominator: \( \frac{2\sqrt{6}}{3} \). This is the expression with a rationalized denominator.

Key Concepts

Understanding Radicals

Radicals are symbols used in mathematics, represented by the square root symbol \( \sqrt{} \), that indicate the root of a number. Specifically, when you see \( \sqrt{9} \), it means you're looking for the number that, when multiplied by itself, gives 9. In this case, the answer is 3, since 3 times 3 equals 9.

Radicals can be more complex when dealing with numbers that aren’t perfect squares, like 8 or 15. If you have \( \sqrt{8} \), you must consider factors of 8 that are perfect squares. In this scenario, you can see that 8 is 4 times 2, and since 4 is a perfect square of 2, \( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \).

It's important to remember:

• Radicals are another way to represent fractions.
• The process often involves identifying perfect squares within the radical.
• Breaking down the numbers inside the radical can simplify the expression significantly.

Understanding radicals is key to simplifying more complex mathematical expressions.

Mastering Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining equivalence. It can include combining like terms, reducing fractions, or, as in our exercise, rationalizing denominators. When you simplify an expression, it becomes easier to interpret and use in calculations.

Simplifying might involve:

• Combining Like Terms: If there are multiple instances of the same variable or radical within an expression, they can often be combined to simplify the expression.
• Factoring: Expressions can often be broken into products of simpler expressions. Factoring can help identify these and reduce the complexity of an expression.
• Cleaning Up Radicals: As shown in the example, reducing the complexity of a square root is a valuable skill for simplification.

In the provided exercise, simplifying involved breaking down \( \sqrt{24} \) to \( 2\sqrt{6} \), ultimately leading to a more refined expression without an irrational denominator. Simplifying expression saves time and makes computations clearer.

Demystifying Square Roots

Square roots are a specific type of radical that are very common in mathematics. Understanding them is essential for rationalizing denominators and many other math problems.

The square root of a number \( n \) is a number \( m \) such that \( m^2 = n \). Commonly used square roots include values like \( \sqrt{25} = 5 \) because \( 5 \times 5 = 25 \).

Important points about square roots:

• Perfect Squares: Numbers like 1, 4, 9, 16, and 25 are perfect squares because they are squares of whole numbers.
• Non-perfect Squares: Numbers like 2, 3, and 10 cannot be neatly broken down into a single number for their square roots. These require approximation or expression in terms of radicals.
• Rationalization: When dealing with fractions that have square roots in the denominator, it's often necessary to "rationalize" the denominator by manipulating the fraction to achieve a rational, non-square root, denominator.

For instance, with \( \sqrt{\frac{8}{3}} \), multiplying numerator and denominator by \( \sqrt{3} \) rationalizes the denominator, turning it into \( 3 \), which is a whole number. This ensures clarity and simplicity in mathematical solutions and further computations.