Compare (√40-√10)²: Determine if Greater or Less Than 30

Fill in the corresponding sign

$(\sqrt{40}-\sqrt{10})^2?30$

To solve this problem, we'll follow these steps:

• Step 1: Simplify the expression $(\sqrt{40} - \sqrt{10})^2$
• Step 2: Calculate the square root values for $\sqrt{40}$ and $\sqrt{10}$
• Step 3: Compare the simplified value to 30

Now, let's work through each step:
Step 1: Simplify $(\sqrt{40} - \sqrt{10})^2$ using the square of a difference formula:

$(\sqrt{40} - \sqrt{10})^2 = (\sqrt{40})^2 - 2 \cdot \sqrt{40} \cdot \sqrt{10} + (\sqrt{10})^2$

$(\sqrt{40})^2 = 40$ and $(\sqrt{10})^2 = 10$, so:

$(\sqrt{40} - \sqrt{10})^2 = 40 - 2 \cdot \sqrt{40} \cdot \sqrt{10} + 10$

Step 2: Calculate each term:
$\sqrt{40} \approx 6.324$ and $\sqrt{10} \approx 3.162$.

The expression becomes:

$40 - 2 \cdot 6.324 \cdot 3.162 + 10 = 50 - 40 \approx 10$

Step 3: Compare $10$ to $30$. Since $10 < 30$, the correct relation is:

Therefore, the solution to the problem is the comparison $(\sqrt{40} - \sqrt{10})^2 < 30$.

Thus, the sign is $\textbf{<}$.