Problem 87 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ -5 \sqrt{2} \cdot \sqrt{\frac{9}{50}} $$

Step-by-Step Solution

Answer

The simplification of the given expression gives \(-3\).

Step 1: Simplify the Fractions

First step is to simplify the expression \(\sqrt{\frac{9}{50}}\). To do so, square root the numerator and denominator separately. \(\sqrt{\frac{9}{50}} = \frac{\sqrt{9}}{\sqrt{50}} = \frac{3}{\sqrt{50}}\)

Step 2: Simplify the radical in denominator

Next step is to simplify \(\sqrt{50}\) which is \(\sqrt{25*2}\). After simplifying, denominator will become \(\frac{3}{5\sqrt{2}}\)

Step 3: Rationalize the denominator

To avoid a radical in denominator, multiply both, numerator and the denominator by the conjugate of the denominator which would be \(\sqrt{2}\). So, the simplified value of \(\sqrt{\frac{9}{50}} = \frac{3\sqrt{2}}{10}\)

Step 4: Multiplication of Radical Terms

According to the original expression, multiply -5√2 and the simplified fraction from the previous step: \(-5 \sqrt{2} * \frac{3\sqrt{2}}{10} = -\frac{30}{10} = -3\)

Key Concepts

Understanding Radical Expressions

Radical expressions are mathematical expressions that involve the radical symbol, better known as the square root symbol. They are essentially expressions that use roots to represent numbers. A good example is \( \sqrt{2} \), which is a simple radical expression.

• Radicals can be expressions involving square roots, cube roots, or higher order roots.
• Understanding how to work with radical expressions is a foundational skill in algebra and is important in many areas of higher mathematics.

To simplify radical expressions, look for numbers under the radical sign that have perfect squares as factors. This allows us to "take out" numbers from the radical sign, making the expression simpler and easier to work with.

Rationalizing the Denominator

Rationalizing the denominator is a process of removing radicals from the denominator of a fraction. This simplification is necessary because having radicals in the denominator can make expressions difficult to work with and understand.

• To rationalize the denominator, multiply both the numerator and the denominator of the fraction by the radical present in the denominator.
• For example, in simplifying \( \frac{3}{\sqrt{50}} \), we multiply by \( \sqrt{2} \), resulting in \( \frac{3\sqrt{2}}{10} \).

The goal is to have a rational number (a number without any radicals) in the denominator, which often results in a more "standardized" expression.

Simplifying Fractions and Expressions

Simplifying fractions and expressions involves reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1. Simplification makes the expression easier to understand and work with.

• In our example, the fraction \( \frac{3}{10} \) has a numerator and denominator that share no common factors, hence it's already simplified.
• Simplifying is crucial for maintaining accuracy and ensuring that comparisons or calculations are based on the simplest possible representation of values.

Through simplifying fractions and other expressions, we create a cleaner, more organized version of the original problem, which is easier to solve or analyze further.