Problem 24 - College Algebra Essentials

Question

Use the quotient rule to simplify the expressions in Exercises \(23-32\) Assume that \(x>0\) $$\sqrt{\frac{1}{49}}$$

Step-by-Step Solution

Answer

The simplified expression for \( \sqrt{\frac{1}{49}} \) is \( \frac{1}{7} \)

Step 1: Recognize the basic radical expression

Recognize the given expression \( \sqrt{\frac{1}{49}} \) as a radical quotient. Here, \(a=1\) and \(b=49\).

Step 2: Apply the quotient rule

Apply the quotient rule, which states that the square root of a quotient is equal to the quotient of the individual square roots. This changes the expression to \( \frac{\sqrt{1}}{\sqrt{49}} \).

Step 3: Simplify each radical individually

The square root of any number is the number that when multiplied by itself equals the original number. Therefore, \( \sqrt{1} = 1 \) and \( \sqrt{49} = 7 \). Substitute these into the expression as follows: \( \frac{\sqrt{1}}{\sqrt{49}} = \frac{1}{7} \).

Step 4: Final Simplified Expression

The expression \( \sqrt{\frac{1}{49}} \) simplifies to \( \frac{1}{7} \).

Key Concepts

Radical Expressions

In algebra, radical expressions are expressions that include a root, such as a square root or cube root. The square root, which is the most common type of root used in mathematics, is denoted by the symbol \( \sqrt{\cdot} \). A radical expression like \( \sqrt{\frac{1}{49}} \) involves taking the square root of a fraction. Understanding the properties of radicals is crucial to simplifying these kinds of expressions.

To effectively work with radical expressions, you should remember a couple of essential rules:

• The Product Rule of Radicals: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).
• The Quotient Rule of Radicals: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), assuming \( b \) is not zero.

Simplifying radicals involves applying these rules and also recognizing perfect squares and perfect cubes within the radical to reduce the expression to its simplest form.

Simplifying Radicals

The process of simplifying radicals boils down to reducing the expression under the radical to its simplest form. This involves identifying and pulling out perfect squares (or higher powers, for cube roots and so on) from under the radical. In our example, \( \sqrt{\frac{1}{49}} \), we identify that both the numerator and the denominator are perfect squares.

The steps to simplify a radical are as follows:

• Identify perfect squares, cubes, etc., in the radical expression.
• Use the quotient rule to separate the numerator and the denominator under individual radicals if you have a fraction.
• Simplify the radicals individually, turning them into integers where possible.
• Combine the simplified radicals to get your final expression.

It's important always to check your final expression to see if it can be simplified further by factoring or reducing any remaining fractions.

Square Roots

The square root of a number \( x \) is a value that, when multiplied by itself, gives the number \( x \). Symbolically, it is represented as \( \sqrt{x} \). Square roots are fundamental in algebra and are especially important when dealing with quadratic equations and Pythagorean Theorem in geometry.

Some key points to remember include:

• The square root of a perfect square (like 1, 4, 9, 16, 25, etc.) is always an integer.
• Negative numbers do not have real number square roots because a negative times a negative results in a positive.

Mastering the concept of square roots is crucial for simplifying radical expressions and solving a wide range of mathematical problems.