Is the square root of 30 a rational number?
Understand the Problem
The question is asking whether the square root of 30 can be expressed as a fraction of two integers. To determine this, we need to analyze the properties of square roots and rational numbers.
Answer
No, $\sqrt{30}$ is irrational.
Answer for screen readers
No, $\sqrt{30}$ cannot be expressed as a fraction of two integers; it is an irrational number.
Steps to Solve
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Identify the Nature of the Square Root First, we examine whether $\sqrt{30}$ can be expressed as a fraction. This is true if it’s a rational number, which means it can be represented as $\frac{p}{q}$, where $p$ and $q$ are integers, and $q \neq 0$.
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Assume it is Rational and Square Both Sides Assume that $\sqrt{30}$ can be expressed as a rational number. If we set $\sqrt{30} = \frac{p}{q}$, squaring both sides gives us: $$ 30 = \frac{p^2}{q^2} $$ This can be rewritten as: $$ p^2 = 30q^2 $$
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Analyze the Prime Factorization Next, we analyze the prime factorization of 30: $$ 30 = 2 \times 3 \times 5 $$ Since 30 is not a perfect square (its prime factors do not have even powers), $p^2$ must also have the same property, indicating the impossibility of finding integers $p$ and $q$ such that both sides hold true.
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Conclude on the Nature of the Root Since we have shown that the equation leads to a contradiction involving the prime factors, it implies that $\sqrt{30}$ cannot be expressed as a fraction of two integers, meaning it is an irrational number.
No, $\sqrt{30}$ cannot be expressed as a fraction of two integers; it is an irrational number.
More Information
The number $\sqrt{30}$ is considered an irrational number because it cannot be precisely expressed as a fraction. This means its decimal form goes on forever without repeating, which is a common characteristic of irrational numbers.
Tips
- Assuming that any non-integer square root can be expressed as a fraction. Remember, you must check the prime factorization to determine if the number under the square root is a perfect square.
