Solve the Square Root Product: √1 × √2 Step-by-Step

Square Root Multiplication with Identity Properties

Solve the following exercise:

1⋅2= \sqrt{1}\cdot\sqrt{2}=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:06 Let's solve this problem together.
00:10 Imagine the square root of a number A, times the square root of another number B.
00:16 This equals the square root of the product, A times B.
00:21 Now, let's use this formula in our exercise and calculate the product.
00:26 That's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve the following exercise:

1⋅2= \sqrt{1}\cdot\sqrt{2}=

2

Step-by-step solution

Let's start by recalling how to define a square root as a power:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}}

Next, we remember that raising 1 to any power always gives us 1, even the half power we got from converting the square root.

In other words:

1⋅2=↓12⋅2=112⋅2=1⋅2=2 \sqrt{1} \cdot \sqrt{2}= \\ \downarrow\\ \sqrt[2]{1}\cdot \sqrt{2}=\\ 1^{\frac{1}{2}} \cdot\sqrt{2} =\\ 1\cdot\sqrt{2}=\\ \boxed{\sqrt{2}} Therefore, the correct answer is answer a.

3

Final Answer

2 \sqrt{2}

Key Points to Remember

Essential concepts to master this topic
  • Identity Rule: 1=1 \sqrt{1} = 1 because 1 raised to any power equals 1
  • Technique: Use 1⋅2=1⋅2=2 \sqrt{1} \cdot \sqrt{2} = 1 \cdot \sqrt{2} = \sqrt{2}
  • Check: Verify (2)2=2 (\sqrt{2})^2 = 2 and 1⋅2=2 1 \cdot 2 = 2 ✓

Common Mistakes

Avoid these frequent errors
  • Multiplying radicands incorrectly
    Don't think 1⋅2=1+2=3 \sqrt{1} \cdot \sqrt{2} = \sqrt{1 + 2} = \sqrt{3} ! This confuses addition with multiplication of square roots. Always remember that a⋅b=a⋅b \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} , not a+b \sqrt{a + b} .

Practice Quiz

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Choose the largest value

FAQ

Everything you need to know about this question

Why does 1=1 \sqrt{1} = 1 ?

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Because 1 multiplied by itself equals 1! The square root asks "what number times itself gives me this result?" Since 1×1=1 1 \times 1 = 1 , we have 1=1 \sqrt{1} = 1 .

Can I use the rule a⋅b=ab \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} here?

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Absolutely! This gives us 1⋅2=1×2=2 \sqrt{1} \cdot \sqrt{2} = \sqrt{1 \times 2} = \sqrt{2} . Both methods work and give the same answer!

What's the difference between 1⋅2 \sqrt{1} \cdot \sqrt{2} and 1+2 \sqrt{1 + 2} ?

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These are completely different! 1⋅2=2 \sqrt{1} \cdot \sqrt{2} = \sqrt{2} (multiplication), but 1+2=3 \sqrt{1 + 2} = \sqrt{3} (addition inside the radical).

How do I know when to multiply square roots?

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When you see multiplication symbols between square roots like a⋅b \sqrt{a} \cdot \sqrt{b} , you can combine them as a⋅b \sqrt{a \cdot b} or evaluate each root first then multiply.

Why is the answer 2 \sqrt{2} and not a decimal?

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Both forms are correct! 2 \sqrt{2} is the exact answer, while 1.414... is the decimal approximation. In math, we often prefer exact answers when possible.

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