Problem 9 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{8}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{4}\).

Step 1: Multiply the Radicals

To simplify the expression, first multiply the two radicals. According to the property of radicals, \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Therefore, \(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{8}} = \sqrt{\frac{1}{2} \cdot \frac{1}{8}}\).

Step 2: Simplify the Product Inside the Radical

Now simplify the fraction inside the square root: \(\frac{1}{2} \cdot \frac{1}{8} = \frac{1 \cdot 1}{2 \cdot 8} = \frac{1}{16}\). Thus, the expression becomes \(\sqrt{\frac{1}{16}}\).

Step 3: Simplify the Square Root

To find \(\sqrt{\frac{1}{16}}\), notice that \(\frac{1}{16}\) is a perfect square, because \((\frac{1}{4})^2 = \frac{1}{16}\). Therefore, \(\sqrt{\frac{1}{16}} = \frac{1}{4}\). The simplified expression is \(\frac{1}{4}\).

Key Concepts

Square Roots

Square roots are fundamental in algebra, allowing us to reverse the operation of squaring a number. When we talk about the square root of a number, we ask, "What number, when multiplied by itself, gives us the original number?" This is written in the form \( \sqrt{a} \), where \( a \) is the number for which we are finding the square root. For example, since \( 4 \times 4 = 16 \), the square root of 16 is 4, written as \( \sqrt{16} = 4 \).
In our exercise, we had to find the square root of a fraction. Importantly, the square root of a fraction \( \frac{a}{b} \) can be separated into two parts: \( \sqrt{a} \) and \( \sqrt{b} \). This rule helps to simplify expressions, especially when fractions are involved. For instance, \( \sqrt{\frac{1}{16}} = \sqrt{\frac{1}{4 \times 4}} = \frac{1}{4} \). Recognizing when a fraction is a perfect square is key to simplifying these types of problems.

Simplifying Expressions

Simplifying expressions in algebra often involves reducing complex fractions and radical terms to their simplest form. The main goal is to make the expression as straightforward as possible. When simplifying an expression like \( \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{8}} \), we use properties of radicals to combine the radicals first, leading us to \( \sqrt{\frac{1}{16}} \). This step involves understanding the property: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
After combining, it is vital to recognize simple arithmetic and properties of fractions. Multiplying \( \frac{1}{2} \) by \( \frac{1}{8} \) is straightforward: multiply the numerators and denominators separately, which gives \( \frac{1 \cdot 1}{2 \cdot 8} = \frac{1}{16} \). Finally, simplifying the square root of a perfect square fraction results in clear and simple outcomes.

Multiplication of Radicals

The multiplication of radicals is a foundational concept that requires understanding the radical product property, which states \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This property simplifies the multiplication of two radical expressions into a single radical expression, which is easier to work with. In the given problem, using this property, \( \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{8}} \) becomes \( \sqrt{\frac{1}{2} \times \frac{1}{8}} \).
When multiplying radicals, especially those with fractions, it simplifies the work into a manageable step like \( \sqrt{\frac{1}{16}} \). The crucial step is the multiplication inside the radical, which leads us to identify whether the result, \( \frac{1}{16} \), is a perfect square, hence allowing further simplification. Understanding the product property helps unlock solutions to seemingly complex radical expressions.