Is the square root of 20 a rational number?
Understand the Problem
The question is asking whether the square root of 20 is a rational number. To determine this, we need to understand the definition of rational numbers and evaluate the square root of 20.
Answer
The square root of 20 is not a rational number.
Answer for screen readers
The square root of 20 is not a rational number.
Steps to Solve
- Definition of Rational Numbers
A rational number is any number that can be expressed as the fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.
- Evaluating the Square Root of 20
To determine if $\sqrt{20}$ is rational, we can simplify it. We break down 20 into its prime factors:
$$ 20 = 4 \times 5 = 2^2 \times 5 $$
The square root can be simplified as follows:
$$ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} $$
- Analyzing Square Root of 5
Now we need to determine if $\sqrt{5}$ is rational. The number 5 is not a perfect square, meaning it cannot be expressed as the square of an integer. Thus, $\sqrt{5}$ is an irrational number.
- Final Evaluation
Since $\sqrt{20}$ simplifies to $2\sqrt{5}$, and $\sqrt{5}$ is irrational, it follows that $2\sqrt{5}$ is also irrational. Therefore, $\sqrt{20}$ is not a rational number.
The square root of 20 is not a rational number.
More Information
The number $\sqrt{20}$ can be expressed as $2\sqrt{5}$, and since $\sqrt{5}$ is known to be irrational, it confirms that $\sqrt{20}$ is also irrational. Rational numbers appear frequently in various mathematical concepts and are essential for different fields, such as algebra and number theory.
Tips
- Assuming that the square root of any non-integer is rational. This is incorrect as many square roots (like $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$) are irrational.
- Miscalculating the factors of a number or failing to simplify the square root fully.
