Understanding Rational and Irrational Square Roots

A rational number is any number that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Examples include 2 ($\frac{2}{1}$), -3.5 ($\frac{-7}{2}$), and $\frac{1}{3}$.

An irrational number is a number that cannot be expressed as a simple fraction. Their decimal representations are non-terminating and non-repeating. Examples include $\pi$ and $\sqrt{2}$.

The square root of an integer is rational if and only if the integer is a perfect square. A perfect square is an integer that is the square of another integer (e.g., 9 is a perfect square because $3^2=9$). If an integer is not a perfect square, its square root is irrational.

To determine which of the given numbers has an irrational square root, we need to find which one is not a perfect square.

Analyzing Each Option for Rational or Irrational Square Root

Option 1: 4489

Let's find the square root of 4489. We can try estimating or using a calculator. Notice that $60^2 = 3600$ and $70^2 = 4900$. The number ends in 9, so its square root might end in 3 or 7. Let's try 67:

$\sqrt{4489} = 67$

Since 67 is an integer, which is a rational number, $\sqrt{4489}$ is rational. Thus, 4489 is a perfect square ($67^2 = 4489$).

Option 2: 7840

Let's find the square root of 7840. We can use prime factorization to determine if it's a perfect square.

Prime factorization of 7840:

• $7840 = 10 \times 784$
• $10 = 2 \times 5$
• $784 = 2 \times 392 = 2 \times 2 \times 196 = 2 \times 2 \times 2 \times 98 = 2 \times 2 \times 2 \times 2 \times 49 = 2^4 \times 7^2$
• So, $7840 = (2 \times 5) \times (2^4 \times 7^2) = 2^{1+4} \times 5^1 \times 7^2 = 2^5 \times 5^1 \times 7^2$

For a number to be a perfect square, all the exponents in its prime factorization must be even. In the prime factorization of 7840 ($2^5 \times 5^1 \times 7^2$), the exponents are 5, 1, and 2. The exponents 5 and 1 are odd.

Therefore, 7840 is not a perfect square. Its square root will be irrational:

$\sqrt{7840} = \sqrt{2^5 \times 5^1 \times 7^2} = \sqrt{2^4 \times 2^1 \times 5^1 \times 7^2} = 2^2 \times 7 \times \sqrt{2^1 \times 5^1} = 4 \times 7 \times \sqrt{10} = 28\sqrt{10}$

Since $\sqrt{10}$ is irrational (as 10 is not a perfect square), $28\sqrt{10}$ is also irrational.

Option 3: 1024

Let's find the square root of 1024. We know that $30^2 = 900$ and $35^2 = 1225$. Let's try numbers between 30 and 35. The number ends in 4, so its square root might end in 2 or 8. Let's try 32:

$32^2 = 1024$

So, $\sqrt{1024} = 32$. Since 32 is an integer (a rational number), $\sqrt{1024}$ is rational. Thus, 1024 is a perfect square.

Option 4: 2916

Let's find the square root of 2916. We know $50^2 = 2500$ and $60^2 = 3600$. The number ends in 6, so its square root might end in 4 or 6. Let's try 54:

$54^2 = 2916$

So, $\sqrt{2916} = 54$. Since 54 is an integer (a rational number), $\sqrt{2916}$ is rational. Thus, 2916 is a perfect square.

Conclusion

Based on our analysis, the square roots of 4489, 1024, and 2916 are rational because these numbers are perfect squares. The number 7840 is not a perfect square, and therefore, its square root is irrational.

Revision Table: Rational and Irrational Numbers

Additional Information: Identifying Perfect Squares and Irrational Roots

• One way to check if a number is a perfect square is by looking at its prime factorization. If every prime factor appears with an even exponent, the number is a perfect square.
• For example, $36 = 2^2 \times 3^2$. Exponents (2, 2) are even, so 36 is a perfect square ($\sqrt{36}=6$).
• $12 = 2^2 \times 3^1$. The exponent of 3 is 1 (odd), so 12 is not a perfect square ($\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$, which is irrational).
• The square root of any positive integer that is not a perfect square is an irrational number.