Problem 70 - Intermediate Algebra

Question

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{4}{81}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{\frac{4}{81}} \) is \( \frac{2}{9} \).

Step 1: Understand the Problem

We are asked to simplify the radical expression \( \sqrt{\frac{4}{81}} \). This involves simplifying the square root of a fraction.

Step 2: Use the Property of Radicals

Recall that \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This property allows us to separate the square root of the numerator and the denominator.

Step 3: Apply the Property to the Radical

Apply the property from Step 2 to get \( \frac{\sqrt{4}}{\sqrt{81}} \). This means we need to find the square roots of 4 and 81 respectively.

Step 4: Simplify the Numerator

The square root of 4 is 2, since 2 multiplied by itself gives 4, i.e. \( \sqrt{4} = 2 \).

Step 5: Simplify the Denominator

The square root of 81 is 9, since 9 multiplied by itself gives 81, i.e. \( \sqrt{81} = 9 \).

Step 6: Write the Simplified Expression

Combine the simplified numerator and denominator to get \( \frac{2}{9} \). Thus, the simplified form of \( \sqrt{\frac{4}{81}} \) is \( \frac{2}{9} \).

Key Concepts

square roots

Square roots are a foundational concept in mathematics that help us determine which number, when multiplied by itself, equals the number under the square root symbol, known as the radicand. For instance, the square root of 4 is 2 because 2 times 2 equals 4. Similarly, for the square root of 81, the result is 9, since 9 times 9 equals 81.

Understanding square roots is essential, especially when working with fractions. When you encounter a square root of a fraction, such as \( \sqrt{\frac{4}{81}} \), it is crucial to apply the properties of radicals to simplify the expression by finding the square roots of the numerator and the denominator separately.

radical expressions

Radical expressions involve roots, such as square roots, cube roots, etc., and often include variables or numbers under the root sign. They are a broader concept in algebra and are used to represent numbers we cannot easily express using arithmetic alone.

The key to simplifying radical expressions, especially those in fraction form like \( \sqrt{\frac{a}{b}} \), is to use the property that allows you to break them into separate roots: \( \frac{\sqrt{a}}{\sqrt{b}} \). This separation makes it easier to simplify each part individually.

• For example, \( \sqrt{\frac{4}{81}} \) becomes \( \frac{\sqrt{4}}{\sqrt{81}} \), where each part can be simplified more straightforwardly by determining the square roots of 4 and 81.
• Don't forget to ensure that all variables involved are positive to avoid dealing with imaginary numbers.

Breaking down radical expressions this way helps manage complexity and achieves a clearer and more concise form.

properties of radicals

Properties of radicals are rules that help us manipulate and simplify radical expressions effectively. These properties make the process of working with roots much more intuitive and manageable. Here are a few important properties to remember when dealing with radicals:

• Multiplication Property: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This means you can break down the square root of a product into the product of square roots.
• Division Property: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This property is helpful when simplifying the square root of a fraction by allowing separate consideration of the numerator and denominator, as we did in the exercise \( \sqrt{\frac{4}{81}} = \frac{2}{9} \).
• Power Property: \( \sqrt{a^n} = a^{n/2} \), which is useful when dealing with expressions involving powers and roots.

Using these properties, you can confidently approach and simplify even complex radical expressions, understanding each step along the way. It builds a strong foundation for more advanced topics in algebra and calculus.