Problem 23 - Algebra 2

Question

Divide and simplify. Assume that all variables are positive. $$ \frac{\sqrt{500}}{\sqrt{5}} $$

Step-by-Step Solution

Answer

The simplified form of \( \frac{\sqrt{500}}{\sqrt{5}} \) is 10.

Step 1: Factor Out the Greatest Perfect Square From 500

Following the laws of square roots, we should start by simplifying the numerator. 500 can be factored into 100 and 5, so the square root of 500 can be written as the square root of (100 * 5). The square root of 100 is 10, so the square root of 500 simplifies to \(10\sqrt{5}\). Our fraction thus becomes \( \frac{10\sqrt{5}}{\sqrt{5}} \).

Step 2: Divide the Radicals

Now, divide out the common term from the numerator and denominator, which is \( \sqrt{5} \). After conducting the division, the fraction simplifies to just 10.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics. They are used to find a number which, when multiplied by itself, results in the original number. When we see the notation \( \sqrt{a} \), it indicates the square root of \( a \). In simple terms, if \( b^2 = a \), then \( b \) is the square root of \( a \).

In our exercise, understanding how to work with square roots is key. We have \( \sqrt{500} \) and \( \sqrt{5} \). By simplifying \( \sqrt{500} \) correctly, we make the process of division much easier. Notice that any square root can potentially be simplified by breaking it down into factors, particularly focusing on perfect squares.

Factoring Perfect Squares

Factoring involves breaking down a number into its prime factors or into a product of other numbers. Perfect squares are numbers that have whole numbers as their square roots, like 4, 9, 16, 25, etc. In our context, identifying perfect squares helps simplify the square roots.

For the number 500, we can factor it as \( 100 \times 5 \). Here, 100 is a perfect square (since \( 10^2 = 100 \)). Recognizing this helps us simplify since \( \sqrt{100} = 10 \). That gives us \( 10\sqrt{5} \) instead of \( \sqrt{500} \).

• Start by finding the largest perfect square that can factor into the given number.
• Use the property: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
• Converting to a simpler form makes further operations, like division, more straightforward.

Algebraic Division

Algebraic division involves dividing expressions that contain variables or radicals, like in our exercise. When dividing radicals, such as \( \frac{\sqrt{a}}{\sqrt{b}} \), remember the property \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). This is especially useful when there's a common factor in the numerator and the denominator.

In our example, dividing \( \frac{10\sqrt{5}}{\sqrt{5}} \) simplifies the expression considerably. Here, the \( \sqrt{5} \) terms cancel out, leaving just 10. Understanding this cancellation process is crucial in algebraic division, particularly when simplifying radical expressions.

Be cautious not to leave any radicals in the denominator (a process called rationalizing the denominator) unless they cancel neatly as they do here.

Mathematical Simplification

Simplification is all about reducing expressions to their simplest form. It not only makes numbers easier to work with but also reduces the potential for errors in further calculations.

To simplify a complex expression, look for ways to reduce, eliminate, or combine terms. In our problem, we first simplified \( \sqrt{500} \) into \( 10\sqrt{5} \) using factorization. Then, by applying algebraic division, we reduced the fraction to just 10.

Simplification in mathematics often includes:

• Removing radicals or fractions where possible.
• Combining like terms and reducing coefficients.
• Eliminating unnecessary terms to make an expression compact and comprehensible.

By mastering these techniques, you'll be able to tackle a broad range of mathematical problems with greater confidence.