Problem 45 - Contemporary Precalculus

Question

Rationalize the denominator and simplify your answer. $$\frac{3}{\sqrt{8}}$$

Step-by-Step Solution

Answer

Answer: The simplified and rationalized form of the fraction \(\frac{3}{\sqrt{8}}\) is \(\frac{3\sqrt{2}}{4}\).

Step 1: Simplify the denominator

To simplify the denominator, first, find the prime factorization of 8: \(2^3\). Now, divide the exponent (3) by 2 to find what can be factored out of the square root. \(3 \div 2 = 1 \text{ remainder } 1\). This shows that one factor of 2 can be factored out from the square root: $$\frac{3}{\sqrt{8}} = \frac{3}{\sqrt{2^2 \cdot 2}} = \frac{3}{2\sqrt{2}}$$

Step 2: Rationalize the denominator

To rationalize the denominator, multiply both the numerator and the denominator by the remaining square root, in this case \(\sqrt{2}\): $$\frac{3}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2 \cdot 2}$$

Step 3: Simplify the fraction

Now simplify the fraction by cancelling out any common factors. In this case, the fraction is already simplified: $$\frac{3\sqrt{2}}{2\cdot2} = \frac{3\sqrt{2}}{4}$$ Therefore, the simplified and rationalized answer is: $$\frac{3}{\sqrt{8}} = \frac{3\sqrt{2}}{4}$$

Key Concepts

Simplifying Radicals

When we talk about simplifying radicals, we refer to the process of breaking down a square root or other radical expressions into their simplest form. This simplification often involves identifying perfect squares that are factors of the number under the square root sign. Simplifying radicals makes the numbers more manageable and prepares them for operations such as addition, subtraction, or further algebraic manipulations.

For instance, in the exercise \frac{3}{\sqrt{8}}\, we recognized that the radical can be simplified. The number 8 can be broken down into 2 x 2 x 2, where 2 x 2 is a perfect square, and its square root can be taken out from under the radical sign. This process enables us to write 8 as \(2^2\cdot2\), simplifying the expression to \(2\sqrt{2}\) when brought out of the radical. This is a crucial step before we can proceed to rationalize the denominator and make the expression cleaner and more straightforward.

Prime Factorization

Prime factorization is the breakdown of a composite number into a product of its prime factors. This method plays an integral role in simplifying radicals, as seen in the provided exercise. To find the prime factors of a number, you can divide it by the smallest prime number possible and continue the process until all factors are prime.

In the exercise with \(\frac{3}{\sqrt{8}}\), we performed prime factorization on the number 8. Since 8 is not a prime number, we looked for prime numbers that multiply to give us 8 and found that 2 x 2 x 2 correctly factors 8. These numbers are all prime factors of 8. With this breakdown, we could identify which factors could be taken out of the radical, which is an essential move towards simplification and ultimately rationalizing the denominator.

Square Roots

A square root is a number that produces a specified quantity when multiplied by itself. Square roots are often found in radical expressions and can sometimes be irrational numbers when they are not perfect squares. For simplifying radicals, recognizing perfect square factors, as with prime factorization, is the key to simplification.

Understanding square roots is also crucial in the process of rationalizing the denominator. By multiplying the denominator by the appropriate square root, as seen with \(\frac{3}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}\), we eliminate the square root from the denominator. This multiplication gives a rational denominator, which is often a necessary condition for a simplified expression, while ensuring that the value of the expression doesn't change because we're effectively multiplying by 1.