Problem 62 - Precalculus: Functions and Graphs

Question

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt{\frac{1}{7}}$$

Step-by-Step Solution

Answer

\(\frac{\sqrt{7}}{7}\)

Step 1: Express the square root of a fraction

Rewrite the expression using the property of square roots: \(\sqrt{\frac{1}{7}} = \frac{\sqrt{1}}{\sqrt{7}}\). This separates the square root of a fraction into a fraction of square roots.

Step 2: Simplify the numerator

Since \(\sqrt{1} = 1\), the expression becomes \(\frac{1}{\sqrt{7}}\). This gives us a fraction with a square root in the denominator.

Step 3: Rationalize the denominator

Multiply both the numerator and the denominator by \(\sqrt{7}\) to eliminate the square root in the denominator: \(\frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{7}}{7}\). This process eliminates the square root from the denominator by turning it into \(7\).

Key Concepts

Square Roots

Square roots are a fundamental mathematical operation used to find a number which, when multiplied by itself, gives the original number. In this exercise, we are dealing with the square root of a fraction, which is a bit trickier.

• The primary rule for square roots is that the square root of a number "x" is a number "y" such that \( y \times y = x \).
• The square root symbol is \( \sqrt{} \), and it is applied to the number underneath it.

When you encounter a square root of a fraction, like \( \sqrt{\frac{1}{7}} \), you can split it into two separate square roots as shown in the original exercise. It becomes \( \frac{\sqrt{1}}{\sqrt{7}} \).By simplifying each part, you handle complex expressions more easily. Here, \( \sqrt{1} \) simplifies to 1 because 1 is a perfect square. This helps transform our problem into a simpler one, \( \frac{1}{\sqrt{7}} \). Square roots often appear in different fields of math, and understanding their properties ensures efficient problem-solving.

Fractional Expressions

Fractional expressions involve fractions, and they can sometimes include exponents, radicals like square roots, or even complex numbers. In our case, we are focused on the fraction \( \frac{1}{\sqrt{7}} \).

• Numerator: This is the top part of the fraction and represents the number of parts taken. In our case, it began as the square root of 1 which simplified to 1.
• Denominator: The bottom part, which tells us the total number of equal parts. Initially, it was \( \sqrt{7} \).

When working with fractional expressions, especially those that involve square roots in the denominator, it is often required to "rationalize the denominator." This means getting rid of the square root in the denominator in favor of a rational number. This requires multiplying by the conjugate, often turning the denominator into a simpler, rationalized form.

Simplifying Expressions

Simplifying expressions is a key skill in math that involves making them as simple as possible without changing their value. When simplifying the expression \( \sqrt{\frac{1}{7}} \), one of the critical steps is ensuring we eliminate square roots from the denominator.Here is the process broken down:

• Rewrite the square root of the fraction as a fraction of square roots, as \( \frac{\sqrt{1}}{\sqrt{7}} \).
• Simplify any square roots that are perfect squares, like \( \sqrt{1} = 1 \).
• Rationalize the Denominator: Multiply both parts of the fraction by \( \sqrt{7} \), removing the square root from the denominator, turning \( \sqrt{7} \times \sqrt{7} \) into 7.
• The final, simplified rational expression becomes \( \frac{\sqrt{7}}{7} \).

Simplifying and rationalizing help make calculations easier and results clearer, especially when used in further mathematical operations.