Solve Square Root Multiplication: Finding the Value of √2 × √2

Q: Why does $ \sqrt{2} \times \sqrt{2} = 2 $ and not $ \sqrt{4} $ ?

Both answers are actually the same! $ \sqrt{4} = 2 $, so you can think of it either way. The key is that when you multiply identical square roots, you get the number that was under the radical.

Q: Does this work for other square roots too?

Yes! This pattern works for any number: $ \sqrt{5} \times \sqrt{5} = 5 $, $ \sqrt{10} \times \sqrt{10} = 10 $, and so on. The square root and squaring are inverse operations.

Q: How do I remember this rule?

Think of it as "square root times itself equals the original number". It's like asking "what number squared gives me 2?" The answer is $ \sqrt{2} $, so $ \sqrt{2} \times \sqrt{2} = 2 $!

Q: Can I use a calculator to check my answer?

Absolutely! Calculate $ \sqrt{2} \approx 1.414 $, then multiply: $ 1.414 \times 1.414 \approx 2 $. This confirms our exact answer of 2.

Square Root Multiplication with Identical Radicals

Solve the following exercise:

$\sqrt{2}\cdot\sqrt{2}=$

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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 together.

00:11 Remember, the square root of A times the square root of B is the square root of A times B.

00:17 So, use this formula in our exercise, then calculate their product.

00:22 First, let's find the square root of 4.

00:25 This is how we find the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt{2}\cdot\sqrt{2}=$

Step-by-step solution

To simplify the given expression, we use two laws of exponents:

A. Defining the root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$ B. The law of multiplying exponents for identical bases:

$(a^m)^n=a^{m\cdot n}$

Let's start from the square root of the exponents using the law shown in A:

$\sqrt{2}\cdot\sqrt{2}= \\ \downarrow\\ 2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}=$ We continue: note that we got a number times itself. According to the definition of the exponent we can write the expression as an exponent of that number. Then- we use the law of exponents shown in B and perform the whole exponent on the term in the parentheses:

$2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}= \\ (2^{\frac{1}{2}})^2=\\ 2^{\frac{1}{2}\cdot2}=\\ 2^1=\\ \boxed{2}$ Therefore, the correct answer is answer B.

Final Answer

$2$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying identical square roots, the result equals the radicand
Technique: Convert $\sqrt{2} \times \sqrt{2}$ to $2^{\frac{1}{2}} \times 2^{\frac{1}{2}} = 2^1 = 2$
Check: Verify that $2 \times 2 = 4$ and $(\sqrt{2})^2 = 2$ ✓

Common Mistakes

Avoid these frequent errors

Adding instead of multiplying square roots
Don't add $\sqrt{2} + \sqrt{2} = 2\sqrt{2}$ when the problem asks for multiplication! This gives approximately 2.83 instead of 2. Always follow the operation sign: multiplication means you can combine identical radicals to get the number under the root.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt{\frac{2}{4}}=$

FAQ

Everything you need to know about this question

Why does $\sqrt{2} \times \sqrt{2} = 2$ and not $\sqrt{4}$ ?

Both answers are actually the same! $\sqrt{4} = 2$ , so you can think of it either way. The key is that when you multiply identical square roots, you get the number that was under the radical.

Does this work for other square roots too?

Yes! This pattern works for any number: $\sqrt{5} \times \sqrt{5} = 5$ , $\sqrt{10} \times \sqrt{10} = 10$ , and so on. The square root and squaring are inverse operations.

What about different square roots like $\sqrt{2} \times \sqrt{3}$ ?

When the numbers under the radicals are different, you multiply them together: $\sqrt{2} \times \sqrt{3} = \sqrt{6}$ . You can only simplify to a whole number when the radicals are identical.

How do I remember this rule?

Think of it as "square root times itself equals the original number". It's like asking "what number squared gives me 2?" The answer is $\sqrt{2}$ , so $\sqrt{2} \times \sqrt{2} = 2$ !

Can I use a calculator to check my answer?

Absolutely! Calculate $\sqrt{2} \approx 1.414$ , then multiply: $1.414 \times 1.414 \approx 2$ . This confirms our exact answer of 2.

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Solve Square Root Multiplication: Finding the Value of √2 × √2

Square Root Multiplication with Identical Radicals

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does $\sqrt{2} \times \sqrt{2} = 2$ and not $\sqrt{4}$ ?

Does this work for other square roots too?

What about different square roots like $\sqrt{2} \times \sqrt{3}$ ?

How do I remember this rule?

Can I use a calculator to check my answer?

🌟 Unlock Your Math Potential

More Questions

Product Property of Square Roots

Solve the Square Root Product: √4 × √4 Step-by-Step

Multiply Square Roots: Calculate √2 × √3

Calculate √5 × √6: Multiplying Square Roots Step-by-Step

Multiply Square Roots: Calculate √3 × √3

Rules of Roots Combined

Multiply Square Roots: Calculate √5 × √10 Step-by-Step

Solve Square Root Multiplication: √16 × √1 Step-by-Step

Solve Square Root Multiplication: √1 × √25

Multiply Square Roots: Calculate √2 × √5 Step-by-Step

Solve Square Root Multiplication: Finding the Value of √2 × √2

Square Root Multiplication with Identical Radicals

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does 2×2=2 \sqrt{2} \times \sqrt{2} = 2 2​×2​=2 and not 4 \sqrt{4} 4​?

Does this work for other square roots too?

What about different square roots like 2×3 \sqrt{2} \times \sqrt{3} 2​×3​?

How do I remember this rule?

Can I use a calculator to check my answer?

🌟 Unlock Your Math Potential

More Questions

Product Property of Square Roots

Solve the Square Root Product: √4 × √4 Step-by-Step

Multiply Square Roots: Calculate √2 × √3

Calculate √5 × √6: Multiplying Square Roots Step-by-Step

Multiply Square Roots: Calculate √3 × √3

Rules of Roots Combined

Multiply Square Roots: Calculate √5 × √10 Step-by-Step

Solve Square Root Multiplication: √16 × √1 Step-by-Step

Solve Square Root Multiplication: √1 × √25

Multiply Square Roots: Calculate √2 × √5 Step-by-Step

Why does $\sqrt{2} \times \sqrt{2} = 2$ and not $\sqrt{4}$ ?

What about different square roots like $\sqrt{2} \times \sqrt{3}$ ?