Problem 45 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \frac{\sqrt{500}}{\sqrt{5}} $$

Step-by-Step Solution

Answer

The expression simplifies to 10.

Step 1: Simplify the Fraction of Square Roots

We start by using the property of square roots that says \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). Here, we have: \[ \frac{\sqrt{500}}{\sqrt{5}} = \sqrt{\frac{500}{5}} = \sqrt{100} \]

Step 2: Evaluate the Square Root

Now, we need to calculate the square root of 100. The square root of 100 is 10 since \( 10 \times 10 = 100 \). Therefore: \[ \sqrt{100} = 10 \]

Key Concepts

Square Roots

The concept of square roots is fundamental in mathematics. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 100 is 10 because multiplying 10 by itself (\(10 \times 10 = 100\)) results in 100. It is important to note that the square root of any positive number is always a positive number or zero. This is because the operation of finding a square root is defined in the realm of positive real numbers and zero.

To simplify square roots, look for perfect square factors within the radicand (the number inside the square root). For example, when simplifying \(\sqrt{500}\), identify that 500 can be broken down into 100 and 5. Since 100 is a perfect square, \(\sqrt{500} = \sqrt{100 \times 5} = \sqrt{100} \times \sqrt{5} = 10\sqrt{5}\). This approach helps simplify complex roots effectively.

Fraction of Square Roots

Working with fractions of square roots can sometimes seem complex, but there's a neat property that simplifies it. When faced with a fraction like \(\frac{\sqrt{a}}{\sqrt{b}}\), it can be simplified using the property \(\sqrt{\frac{a}{b}}\). This property allows you to combine the numerators and denominators inside a single square root, which can simplify calculations significantly.

Consider the given exercise: \(\frac{\sqrt{500}}{\sqrt{5}}\). Applying the property gives \(\sqrt{\frac{500}{5}}\), which simplifies to \(\sqrt{100}\). This transformation simplifies our task further as we only need to evaluate a single, possibly more manageable square root. The key takeaway here is that simplifying a fraction of square roots often reduces to performing simpler arithmetic operations inside a single square root.

Positive Real Numbers

In mathematics, positive real numbers are an extensive set that includes all the numbers you can plot on a number line to the right of zero. This set includes both whole numbers and decimals that are greater than zero. Positive real numbers do not include negatives or zero—keeping operations like square roots within these ensure results are practical and intuitive.

When simplifying expressions involving square roots, it is often stipulated that variables represent positive real numbers. This is to keep operations well-defined and to avoid complications that arise when considering other types of numbers, like imaginary or negative ones. For instance, in our exercise, the assumption that variables are positive real numbers assures us that every step of our simplification—such as working with \(\sqrt{500}\)and simplified versions like \(10\). will yield positive real results, keeping all calculations in the realm of real-world applicable mathematics.