Calculate the Difference: √400 - √225 Step by Step

Q: How do I know if a number is a perfect square?

A number is a perfect square if it equals some integer multiplied by itself. Try counting by squares: 1, 4, 9, 16, 25... or think what number times itself gives me this result?

Q: What if I don't recognize 400 and 225 as perfect squares?

You can use prime factorization! For 400: 400 = 4 × 100 = 2² × 10² = (2×10)² = 20². For 225: 225 = 9 × 25 = 3² × 5² = (3×5)² = 15².

Q: What's the difference between √(400-225) and √400 - √225?

Order of operations matters! √(400-225) means subtract first, then take the square root. But √400 - √225 means take each square root first, then subtract. These give completely different answers!

Q: Why do we get such a small answer from big numbers?

Square roots make numbers smaller! When you subtract two large square roots (20 and 15), the difference can be quite small. That's normal and expected in these problems.

Perfect Square Roots with Subtraction

$\sqrt{400}-\sqrt{225}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Square root of any number (X) squared, root cancels square

00:07 We break down 400 to 20 squared

00:15 We break down 225 to 15 squared

00:19 We'll use this formula in our exercise

00:22 Root and square cancel out

00:28 Let's calculate and solve

00:34 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\sqrt{400}-\sqrt{225}=$

Step-by-step solution

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

Step 1: Find $\sqrt{400}$ .
Step 2: Find $\sqrt{225}$ .
Step 3: Subtract $\sqrt{225}$ from $\sqrt{400}$ .

Now, let's work through each step:
Step 1: Calculate $\sqrt{400}$ . Since $400 = 20^2$ , it follows that $\sqrt{400} = 20$ .

Step 2: Calculate $\sqrt{225}$ . Since $225 = 15^2$ , it follows that $\sqrt{225} = 15$ .

Step 3: Subtract $\sqrt{225}$ from $\sqrt{400}$ :

$20 - 15 = 5$ .

Therefore, the solution to the problem is $\boxed{5}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Perfect Squares: Recognize 400 = 20² and 225 = 15²
Technique: Find each square root first: √400 = 20, √225 = 15
Check: Verify 20² = 400 and 15² = 225, then 20 - 15 = 5 ✓

Common Mistakes

Avoid these frequent errors

Trying to subtract under one square root
Don't write √(400-225) = √175 ≈ 13.2! This changes the entire problem structure and gives a completely wrong answer. Always find each square root separately first, then subtract: √400 - √225 = 20 - 15 = 5.

Practice Quiz

Test your knowledge with interactive questions

$\sqrt{100}=$

FAQ

Everything you need to know about this question

How do I know if a number is a perfect square?

A number is a perfect square if it equals some integer multiplied by itself. Try counting by squares: 1, 4, 9, 16, 25... or think what number times itself gives me this result?

What if I don't recognize 400 and 225 as perfect squares?

You can use prime factorization! For 400: 400 = 4 × 100 = 2² × 10² = (2×10)² = 20². For 225: 225 = 9 × 25 = 3² × 5² = (3×5)² = 15².

Can I use a calculator for square roots?

Yes, but try to recognize perfect squares first! It's faster and helps build your number sense. Use calculators to verify your mental math: √400 should give exactly 20.

What's the difference between √(400-225) and √400 - √225?

Order of operations matters! √(400-225) means subtract first, then take the square root. But √400 - √225 means take each square root first, then subtract. These give completely different answers!

Why do we get such a small answer from big numbers?

Square roots make numbers smaller! When you subtract two large square roots (20 and 15), the difference can be quite small. That's normal and expected in these problems.

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