To determine whether the expression 2√3 + 2√6 is rational or not, we first need to understand what rational numbers are. A rational number can be expressed as the quotient of two integers, where the denominator is not zero.
Let’s break down the expression:
- 2√3: This part represents 2 times the square root of 3. The square root of 3 is an irrational number, which means it cannot be expressed as a fraction of two integers.
- 2√6: Similarly, this part represents 2 times the square root of 6, which is also an irrational number.
When you add two irrational numbers, the result can be either rational or irrational, depending on the specific numbers involved. In this case, we can simplify the expression:
To analyze further, let’s factor out the common factor:
2√3 + 2√6 = 2(√3 + √6)
Now, we need to analyze the sum (√3 + √6). Both √3 and √6 are irrational. The sum of two distinct irrational numbers is typically irrational as well.
Thus, √3 + √6 remains irrational, and consequently:
2(√3 + √6) is also irrational.
In conclusion, the expression 2√3 + 2√6 is not a rational number.