Problem 46 - Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \frac{\sqrt{128}}{\sqrt{2}} $$

Step-by-Step Solution

Answer

The simplified expression is 8.

Step 1: Apply the Property of Radicals

Recall that \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\). Use this property to rewrite the expression: \[\frac{\sqrt{128}}{\sqrt{2}} = \sqrt{\frac{128}{2}}\].

Step 2: Simplify the Fraction

Divide 128 by 2 to simplify the fraction inside the square root: \[\frac{128}{2} = 64\].

Step 3: Evaluate the Square Root

Now, simplify \(\sqrt{64}\). Recognize that 64 is a perfect square, as \(8 \times 8 = 64\). Thus, \(\sqrt{64} = 8\).

Key Concepts

Properties of Radicals

Radicals have fascinating properties that make them easier to manipulate. These properties help in simplifying complex expressions. One key property is the quotient property of radicals. This property states that the square root of a division equals the division of the square roots:

• \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \)

This property comes in handy when simplifying expressions involving radicals. By using this
property, we can transform a division under a radical into a single radical term, which is often
easier to simplify.

Perfect Squares

Perfect squares are numbers whose square roots are whole numbers. Understanding
perfect squares is crucial when simplifying radicals. Some examples of perfect squares
include:

• \( 1^2 = 1 \)
• \( 2^2 = 4 \)
• \( 3^2 = 9 \)
• \( 4^2 = 16 \)
• \( 5^2 = 25 \)
• \( 8^2 = 64 \)

Recognizing a perfect square within a radical allows us to simplify the expression easily.
For instance, when we encounter \( \sqrt{64} \), we know that it simplifies directly to 8,
because 64 is a perfect square

(\( 8 \times 8 = 64 \)).

Square Root Simplification

When simplifying square roots, the goal is to express the radical in its simplest form.
There are a few steps to follow that make this process straightforward.

• **Identify any perfect squares.** Check if the number under the square root is a perfect
  square; if so, simplify it directly.
• **Use properties of radicals.** Apply radical properties such as the quotient property to
  rewrite the expression.
• **Simplify the expression.** Once the expression is rewritten with any recognizable
  patterns, simplify fully to obtain the simplest form.

For example, simplifying \( \frac{\sqrt{128}}{\sqrt{2}} \) involves using the quotient property
to rewrite it as \( \sqrt{\frac{128}{2}} \), reducing inside the radical to \( \sqrt{64} \),
and recognizing 64 as a perfect square. Thus, the result simplifies to 8.