Problem 60 - Intermediate Algebra

Question

Rationalize each denominator. $$ \frac{8}{\sqrt{10}} $$

Step-by-Step Solution

Answer

The rationalized form of \( \frac{8}{\sqrt{10}} \) is \( \frac{4 \sqrt{10}}{5} \).

Step 1: Identify the Rationalization Process

To rationalize the denominator of a fraction involving a square root, multiply the numerator and the denominator by the same square root. Here, we need to remove the square root in the denominator \( \sqrt{10} \) by multiplying both the numerator and denominator by \( \sqrt{10} \).

Step 2: Multiply Numerator and Denominator

Multiply both the numerator and the denominator of the fraction \( \frac{8}{\sqrt{10}} \) by \( \sqrt{10} \). This gives: \[ \frac{8 \cdot \sqrt{10}}{\sqrt{10} \cdot \sqrt{10}} = \frac{8 \sqrt{10}}{10}. \]

Step 3: Simplify the Fraction

The expression \( \frac{8 \sqrt{10}}{10} \) can be simplified since 8 and 10 have a common factor of 2. Divide the numerator and the denominator by 2: \[ \frac{8 \sqrt{10}}{10} = \frac{4 \sqrt{10}}{5}. \] This is the simplest form of the fraction with a rationalized denominator.

Key Concepts

Understanding Square Roots

Square roots are fundamental concepts in mathematics that help us find a number which, when multiplied by itself, gives the original number under the square root symbol. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.
Understanding square roots is crucial when working with irrational numbers. An irrational number is a number that cannot be expressed as a simple fraction, like the square root of 10. While numbers like these cannot be simplified into a whole number or a neat fraction, they are still manageable in mathematical expressions and need to be handled carefully, especially in denominators. This is where the process of rationalizing the denominator comes into play.

• The symbol for square root is "√", and it's applied directly to the number you want to find it for.
• When dealing with square roots, it's essential to remember that \( \sqrt{a^2} = a \).
• Square roots can be found for perfect squares, like 4, 9, 16, and so on, resulting in whole numbers.

Fraction Simplification

Simplifying fractions is a core skill in mathematics that involves reducing fractions to their simplest form. This means making the numerator and the denominator as small as possible while maintaining the same value.
In our problem, fraction simplification occurred after multiplying by the square root. We got \( \frac{8\sqrt{10}}{10} \), which was then simplified by finding a common factor between the numerator and the denominator, in this case, the number 2.
The steps for simplifying fractions include:

• Identify the greatest common factor (GCF) of both numerator and denominator.
• Divide both the numerator and the denominator by this GCF.
• In our example, the GCF of 8 and 10 is 2, hence \( \frac{8\sqrt{10}}{10} = \frac{4\sqrt{10}}{5} \).

Simplifying fractions not only makes calculations easier but is also a common practice in making results more readable in mathematics education.

Role of Mathematics Education

Mathematics education forms the basis for understanding and working with concepts like square roots and fraction simplification. It's about teaching students how to approach and solve problems using logical reasoning and arithmetic skills.
Learning to rationalize the denominator is a key component of this education. It not only helps students understand how to handle irrational numbers but also boosts confidence in manipulating expressions.

• Mathematics educators focus on building foundational skills necessary for understanding more complex algebraic concepts.
• Students are encouraged to practice these problems regularly to build fluency and confidence in their mathematical abilities.

As students progress through mathematics education, they learn to apply these basic concepts in various real-life situations and advanced mathematical theories, laying a vital groundwork for future studies and applications.