Simplifying Expressions with Roots: Rational vs. Irrational Numbers

When dealing with mathematical expressions involving roots (square roots, cube roots, etc.), it is important to understand whether the simplified result is a rational or an irrational number. Understanding this concept not only helps in simplifying complex expressions but also enhances problem-solving skills in mathematics. In this article, we will explore the process of simplifying the expression 2√45 3√20/2√5 and determine whether the final value is rational or irrational.

Step-by-Step Simplification

The given expression is:

2√45 3√20/2√5

To simplify this expression, we will break it down into manageable steps:

Step 1: Simplify 2√45

First, we need to simplify the term 2√45.

2√45  2√(9×5)  2(√9×√5)  2(3√5)  6√5

So, 2√45 simplifies to 6√5.

Step 2: Simplify 3√20/2√5

Next, we will simplify the fraction 3√20/2√5.

3√20  3√(4×5)  3(√4×√5)  3(2√5)  6√5

Thus, the numerator (3√20) simplifies to 6√5, and the denominator (2√5) remains 2√5.

Now, the fraction simplifies to:

3√20/2√5  6√5/2√5  6/2  3

This gives us the simplified value of 3.

Step 3: Combine the Terms

Now, we need to combine the simplified terms:

6√5  3

This results in:

6√5   3

Step 4: Analyze Rationality

To determine whether the final expression is rational or irrational, we need to consider the nature of the components:

6√5 is irrational because √5 is irrational. The rational number 3 is also a valid component.

The sum of a rational number and an irrational number is always irrational. Therefore, the expression 6√5 3 is irrational.

Conclusion:

The expression 2√45 3√20/2√5 simplifies to an irrational number.

Further Examples and Discussion

Let's consider another example to solidify our understanding:

Example: If we were given the expression 2√45 3√20/2√5, we would follow the same steps:

y  2√45  3√20/2√5y  {2√5 × 9  3√5 × 4}/2√5y  6√5  6√5/2√5y  6√5  6/2√5y  6

After simplification, we get 6, which is a rational number.

Another example is the expression 2√45 3√20/2√5 (reiterated):

2√45 3√20/2√5 2√5 √9 3√5 √4/2√5

6√5 6√5/2√5

6√5 3

The final expression 6√5 3 is:

Rational if the expression simplifies to a rational number. Irrational if the expression simplifies to an irrational number.

In all cases, we have demonstrated the process of simplifying expressions involving square roots and analyzing their rationality. Understanding these concepts is crucial for advanced mathematical problem-solving and for ensuring the accuracy of calculations in various fields such as engineering, physics, and finance.