Which of the following numbers is irrational? A) $\sqrt{3}$ B) $\sqrt{144}$ C) $\sqrt{\frac{16}{25}}$ D) $\sqrt{2}+\sqrt{2}$
Understand the Problem
The question asks us to identify an irrational number from a list of options. An irrational number cannot be expressed as a fraction of two integers, and its decimal representation is non-terminating and non-repeating. We need to evaluate each option to determine if it can be simplified to a rational number.
Answer
$\sqrt{5}$
Answer for screen readers
$\sqrt{5}$
Steps to Solve
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Evaluate $\sqrt{4}$ $\sqrt{4}$ is the square root of 4, which is 2. Since 2 can be written as $\frac{2}{1}$, it is a rational number.
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Evaluate $\sqrt{5}$ $\sqrt{5}$ is the square root of 5. 5 is not a perfect square, so its square root is a non-terminating, non-repeating decimal. Therefore, $\sqrt{5}$ is an irrational number.
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Evaluate $\sqrt{9}$ $\sqrt{9}$ is the square root of 9, which is 3. Since 3 can be written as $\frac{3}{1}$, it is a rational number.
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Evaluate $\sqrt{16}$ $\sqrt{16}$ is the square root of 16, which is 4. Since 4 can be written as $\frac{4}{1}$, it is a rational number.
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Identify the irrational number From the evaluations above, $\sqrt{5}$ is the only irrational number.
$\sqrt{5}$
More Information
Irrational numbers are numbers that cannot be expressed as a ratio of two integers. Their decimal representations neither terminate nor repeat. Famous examples of irrational numbers include $\pi$ and $e$.
Tips
A common mistake is to assume that any number under a square root is irrational. However, if the number is a perfect square (like 4, 9, or 16), then its square root is an integer, which is a rational number.
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