Problem 48 - Algebra and Trigonometry

Question

Rationalize the denominator. $$ \frac{\sqrt{7}}{\sqrt{3}} $$

Step-by-Step Solution

Answer

The rationalized form of \( \frac{\sqrt{7}}{\sqrt{3}} \) is \( \frac{\sqrt{21}}{3} \).

Step 1: Identify the problem

The goal is to rationalize the denominator, meaning the square root should not be in the denominator. The given fraction is \( \frac{\sqrt{7}}{\sqrt{3}} \). So, the problem requires to move \( \sqrt{3} \) from the denominator to the numerator.

Step 2: Multiply by a form of one

To achieve the goal, multiply the fraction by a form of '1' that will eliminate the square root from the denominator. Here '1' is in the form of \( \sqrt{3}/\sqrt{3} \). When multiplying the original fraction by this form of '1', it gives \( \frac{\sqrt{7}*\sqrt{3}}{\sqrt{3}*\sqrt{3}} \).

Step 3: Simplify the result

This step involves simplifying the result from step 2. The multiplication \( \sqrt{7}*\sqrt{3} \) gives \( \sqrt{21} \) and the multiplication \( \sqrt{3}*\sqrt{3} \) gives 3. Hence, the rationalized fraction is \( \frac{\sqrt{21}}{3} \).

Key Concepts

Understanding Square Roots

A square root is a number that, when multiplied by itself, produces the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9. We denote square roots using the symbol \( \sqrt{} \), which is called the radical sign. For instance, \( \sqrt{16} = 4 \) because 4 times 4 equals 16. Square roots can also apply to expressions that aren't perfect squares, like \( \sqrt{7} \). This introduces irrational numbers, which are numbers that cannot be expressed as fraction of two integers. To handle such cases, we often rationalize the denominator in mathematical expressions for simplicity and neatness.

Simplifying Fractions

Fractions consist of a numerator (top part) and a denominator (bottom part). Simplifying a fraction involves reducing it to its simplest form. This means making the numerator and denominator as small as possible while maintaining the same value. To simplify, you can divide both by their greatest common divisor (GCD). Sometimes, we need to address irrational numbers in the denominator, such as square roots, as seen in our original exercise problem. Here's what you can do:

• Eliminate the square root from the denominator by multiplying by "1" in the form of the denominator's square root over itself
• Apply multiplication rules for radicals to simplify further
• Rewrite the expression with a rational number in the denominator

The result is a simpler or more conventional representation of the fraction.

Multiplication of Radicals

Multiplying radicals involves some basic rules that make the process straightforward. When you multiply two square roots, such as \( \sqrt{a} \) and \( \sqrt{b} \), you multiply the numbers under the radicals. It's simple: \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). This is because the square root of a product equals the product of the square roots. It becomes handy in expressions like our original problem where \( \sqrt{7} \times \sqrt{3} = \sqrt{21} \). Make sure roots are combined properly and simplified once combined. Remember:

• Keep the multiplication process confined to numbers under the same type of radical
• Simplify after multiplication if possible to express as a reduced radical or a whole number

This multiplication approach ensures expressions are simplified neatly and logically.