Solve Square Root Multiplication: √1 × √25

Square Root Multiplication with Perfect Squares

Solve the following exercise:

125= \sqrt{1}\cdot\sqrt{25}=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:07 Let's solve this problem step by step.
00:10 The square root of A times the square root of B equals the square root of A times B.
00:17 We'll apply this formula to our problem and find the answer.
00:22 Break down 25 as 5 to the power of 2.
00:26 Taking the square root of 5 squared just gives us 5.
00:31 And that's the solution! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve the following exercise:

125= \sqrt{1}\cdot\sqrt{25}=

2

Step-by-step solution

To solve the expression 125 \sqrt{1} \cdot \sqrt{25} , we will use the Product Property of Square Roots.

According to the property, we have:

125=125\sqrt{1} \cdot \sqrt{25} = \sqrt{1 \cdot 25}

First, calculate the product inside the square root:

125=251 \cdot 25 = 25

Now the expression simplifies to:

25\sqrt{25}

Finding the square root of 25 gives us:

55

Thus, the value of 125 \sqrt{1} \cdot \sqrt{25} is 5\boxed{5}.

After comparing this solution with the provided choices, we see that the correct answer is choice 3.

3

Final Answer

5 5

Key Points to Remember

Essential concepts to master this topic
  • Product Property: ab=ab \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} for non-negative values
  • Technique: Multiply inside: 125=125=25 \sqrt{1} \cdot \sqrt{25} = \sqrt{1 \cdot 25} = \sqrt{25}
  • Check: Verify perfect squares: 25=5 \sqrt{25} = 5 because 52=25 5^2 = 25

Common Mistakes

Avoid these frequent errors
  • Multiplying the square root values directly without using the product property
    Don't calculate 1=1 \sqrt{1} = 1 and 25=5 \sqrt{25} = 5 first, then multiply 1 × 5 = 5! While this gives the right answer here, it's the wrong method and will fail with non-perfect squares. Always use the product property: ab=ab \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} first.

Practice Quiz

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Choose the largest value

FAQ

Everything you need to know about this question

Why can't I just find each square root first and then multiply?

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While that works for perfect squares like this problem, it won't work for expressions like 28 \sqrt{2} \cdot \sqrt{8} . The product property ab=ab \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} is the universal method that always works!

What if the numbers under the square roots aren't perfect squares?

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Use the same product property! For example: 28=16=4 \sqrt{2} \cdot \sqrt{8} = \sqrt{16} = 4 . The product inside might become a perfect square even when the original numbers aren't.

Does the order matter when multiplying square roots?

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No! Square root multiplication is commutative, so 125=251 \sqrt{1} \cdot \sqrt{25} = \sqrt{25} \cdot \sqrt{1} . You can multiply in any order.

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

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A perfect square is a whole number that equals another whole number squared. Common ones to memorize: 1=12,4=22,9=32,16=42,25=52,36=62 1=1^2, 4=2^2, 9=3^2, 16=4^2, 25=5^2, 36=6^2 , etc.

What happens if I have three or more square roots to multiply?

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The same rule applies! abc=abc \sqrt{a} \cdot \sqrt{b} \cdot \sqrt{c} = \sqrt{a \cdot b \cdot c} . Just multiply all the numbers under one big square root.

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