Problem 46 - Elementary and 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: Write the Expression as a Single Square Root

Use the property that \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \) to combine the square roots into one. Thus, the expression becomes \( \sqrt{\frac{128}{2}} \).

Step 2: Simplify Inside the Square Root

Divide 128 by 2 to simplify the expression inside the square root: \( \frac{128}{2} = 64 \). Therefore, the expression is now \( \sqrt{64} \).

Step 3: Find the Square Root

Calculate the square root of 64, which is a perfect square. \( \sqrt{64} = 8 \). So the expression simplifies to 8.

Key Concepts

Square Roots

A square root is simply the opposite of squaring a number. If you have a number, say 49, the square root of 49 is 7, because when you multiply 7 by itself, you get 49 again. The symbol for square roots is \( \sqrt{} \). This means that when you want to find the square root of a number, you are searching for the value that, when squared, gives you the original number.

Let's look at the expression: \( \sqrt{128} \). This means you are trying to find a number which, when multiplied by itself, equals 128. When dealing with square roots, it can often be useful to break down the original number into smaller parts, often focusing on perfect squares.

Properties of Radicals

Radicals involve roots of a number, and they come with several useful properties that help in simplifying expressions. One such property is that you can combine roots by division or multiplication of the numbers inside them.

For example, if you have \( \frac{\sqrt{a}}{\sqrt{b}} \), you can combine these into a single square root: \( \sqrt{\frac{a}{b}} \). This property is particularly helpful when you are handling expressions involving fractional roots. In the exercise, converting the expression \( \frac{\sqrt{128}}{\sqrt{2}} \) to \( \sqrt{\frac{128}{2}} \) simplifies the problem by reducing the division inside the root.

Perfect Squares

Perfect squares are numbers that have a whole number as their square root. Examples include 1, 4, 9, 16, 25, and so on. The significance of perfect squares in root calculus is that they tend to simplify expressions easily.

In our exercise, once we divide inside the square root, we get \( \sqrt{64} \). Since 64 is a perfect square (because \( 8 \times 8 = 64 \)), it simplifies directly to a whole number: 8. Recognizing and working with perfect squares can make the process of simplifying radicals much quicker and easier, transforming complex expressions into simpler ones with fewer steps.