Problem 58 - Intermediate Algebra

Question

Rationalize each denominator. $$ \sqrt{\frac{8}{7}} $$

Step-by-Step Solution

Answer

Rationalized expression is \(\frac{2\sqrt{14}}{7}\).

Step 1: Express as one fraction

The expression given is already in the form of \[\sqrt{\frac{8}{7}} = \frac{\sqrt{8}}{\sqrt{7}}.\]Our goal is to rationalize the denominator \(\sqrt{7}\).

Step 2: Multiply by the Rationalizing Factor

To remove the square root from the denominator, multiply both the numerator and the denominator by \(\sqrt{7}\):\[\frac{\sqrt{8}}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{8} \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} = \frac{\sqrt{56}}{7}.\]

Step 3: Simplify the Numerator

Now simplify \(\sqrt{56}\) in the numerator. Notice that \(56 = 4 \times 14 = 4 \times 2 \times 7\):\[\sqrt{56} = \sqrt{4 \times 2 \times 7} = \sqrt{4} \cdot \sqrt{2 \times 7} = 2\sqrt{14}.\]So the expression becomes:\[\frac{2\sqrt{14}}{7}.\]

Key Concepts

Square Roots

When working with square roots, we deal with numbers that, when multiplied by themselves, yield the original number under the root. In simpler terms, the square root of a number is another number that, when squared, gives the first number. For example, the square root of 9 is 3 because 3 squared is 9. Square roots are denoted by the radical symbol \( \sqrt{} \).

In math, especially when dealing with fractions under a square root, it's often necessary to simplify or manipulate these numbers to make equations easier to handle. The goal is to have clear solutions without complex roots at improper places, such as a denominator. Understanding square roots allows you to rationalize denominators, making it easier to work with expressions where roots are involved.

Simplifying Radicals

Simplifying radicals involves breaking down the radicand (the number inside the radical sign) into its simplest form. This is typically done by identifying and extracting perfect square factors.

For example, in the term \( \sqrt{56} \), we recognize that 56 can be factored into \( 4 \times 14 \) and since 4 is a perfect square (as \( \sqrt{4} = 2 \)), it can be taken out of the radical sign. This simplifies \( \sqrt{56} \) to \( 2\sqrt{14} \) as seen in our original solution.

Steps to simplify radicals include:

• Breaking down the radicand into factors.
• Identifying and extracting perfect squares.
• Reducing the expression to its simplest form.

Mastering this process can make complex calculations manageable and keep expressions neat.

Numerator and Denominator Manipulation

In the context of rationalizing denominators, the manipulation of both numerator and denominator is key. This ensures that there are no radicals in the denominator, which is often a preferred form in mathematical expressions.

To do this, we multiply both the numerator and the denominator by the same value, often the radical in the denominator itself. In our exercise, we multiplied \( \frac{\sqrt{8}}{\sqrt{7}} \) by \( \frac{\sqrt{7}}{\sqrt{7}} \), achieving a denominator without a radical since \( \sqrt{7} \times \sqrt{7} = 7 \). This results in an expression where the root is only in the numerator, which is more standard and easier to interpret.

This manipulation involves:

• Identifying the root in the denominator.
• Multiplying by a rationalizing factor.
• Simplifying the resulting expression.

Through these steps, you'll achieve greater clarity and simplicity in your equations.