Is the square root of 50 rational or irrational?
Understand the Problem
The question is asking whether the square root of 50 is a rational or irrational number. To determine this, we need to analyze the number and check if it can be expressed as a fraction of two integers.
Answer
The square root of 50 is an irrational number.
Answer for screen readers
The square root of 50 is an irrational number.
Steps to Solve
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Identify the square root of 50 To determine if $\sqrt{50}$ is rational or irrational, we first need to calculate the square root.
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Simplify the square root We can simplify $\sqrt{50}$ by factoring it: $$ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2} $$
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Determine if $\sqrt{2}$ is rational or irrational The number $\sqrt{2}$ is a well-known irrational number. Thus, since $5$ is a rational number and we are multiplying it with $\sqrt{2}$, the product $5 \sqrt{2}$ remains irrational.
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Conclude about the rationality of $\sqrt{50}$ Since we established that $\sqrt{50} = 5 \sqrt{2}$, and we know that $\sqrt{2}$ is irrational, we conclude that $\sqrt{50}$ is also irrational.
The square root of 50 is an irrational number.
More Information
The number $\sqrt{50}$ can be expressed in its simplest form as $5\sqrt{2}$, which is composed of a rational number ($5$) and an irrational number ($\sqrt{2}$). Since any product that includes an irrational number is itself irrational, this confirms that $\sqrt{50}$ is not a rational number.
Tips
Common mistakes include thinking that because $50$ is a rational number (it can be expressed as $\frac{50}{1}$), its square root must also be rational. However, this is not true for all numbers, as evidenced by $\sqrt{2}$.
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