Multiply Square Roots: Solving √8 × √8

Q: Why does √8 × √8 equal 8 and not √64?

Because when you multiply identical square roots, they "cancel out"! Think of it this way: $ \sqrt{8} \times \sqrt{8} $ means "what number times itself gives 8?" The answer is 8.

Q: Can I use the property √a × √b = √(a×b) here?

Yes, but it's the long way! You'd get $ \sqrt{8} \times \sqrt{8} = \sqrt{8 \times 8} = \sqrt{64} = 8 $. The shortcut is knowing that $ \sqrt{a} \times \sqrt{a} = a $.

Q: Does this rule work for any number under the square root?

Absolutely! $ \sqrt{5} \times \sqrt{5} = 5 $, $ \sqrt{12} \times \sqrt{12} = 12 $, and so on. When you multiply a square root by itself, you always get the original number under the radical.

Q: How do I remember this rule?

Think of square roots as "asking a question": $ \sqrt{8} $ asks "what number squared gives 8?" When you multiply $ \sqrt{8} \times \sqrt{8} $, you're answering that question - the answer is 8!

Q: What if I need to multiply different square roots like √8 × √2?

Then you do use the property $ \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} $. So $ \sqrt{8} \times \sqrt{2} = \sqrt{16} = 4 $. The "shortcut rule" only applies when multiplying identical square roots.

Square Root Multiplication with Same Radicands

Solve the following exercise:

$\sqrt{8}\cdot\sqrt{8}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's solve this problem together!

00:11 Imagine the square root of A times the square root of B.

00:15 This is equal to the square root of A times B.

00:19 Use this formula to calculate the product now.

00:22 Remember, squaring a number and taking its square root cancels each other.

00:27 And that's the solution! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt{8}\cdot\sqrt{8}=$

Step-by-step solution

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$ b. The law of exponents for power of a power:

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

Let's start by converting the square roots to exponents using the law mentioned in a.:

$\sqrt{8}\cdot\sqrt{8}= \\ \downarrow\\ 8^{\frac{1}{2}}\cdot8^{\frac{1}{2}}=$ Let's continue, notice that we got a number multiplied by itself, therefore, according to the definition of exponents we can write the expression we got as a power of that same number, then - we'll use the law of exponents mentioned in b. and perform the exponentiation on the term in parentheses:

$8^{\frac{1}{2}}\cdot8^{\frac{1}{2}}= \\ (8^{\frac{1}{2}})^2=\\ 8^{\frac{1}{2}\cdot2}=\\ 8^1=\\ \boxed{8}$ Therefore, the correct answer is answer c.

Final Answer

$8$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying square roots of same number, result equals that number
Technique: Convert to exponents: $\sqrt{8} \cdot \sqrt{8} = 8^{1/2} \cdot 8^{1/2} = 8^1 = 8$
Check: Verify that $\sqrt{8} \approx 2.83$ and $2.83 \times 2.83 = 8$ ✓

Common Mistakes

Avoid these frequent errors

Adding radicands instead of using multiplication rule
Don't add the numbers under the radicals like √8 + √8 = √16! This gives 4 instead of 8. Square root multiplication follows the rule √a × √a = a. Always remember that multiplying identical square roots equals the radicand itself.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that is equal to the following:

$\sqrt{a}\cdot\sqrt{b}$

FAQ

Everything you need to know about this question

Why does √8 × √8 equal 8 and not √64?

Because when you multiply identical square roots, they "cancel out"! Think of it this way: $\sqrt{8} \times \sqrt{8}$ means "what number times itself gives 8?" The answer is 8.

Can I use the property √a × √b = √(a×b) here?

Yes, but it's the long way! You'd get $\sqrt{8} \times \sqrt{8} = \sqrt{8 \times 8} = \sqrt{64} = 8$ . The shortcut is knowing that $\sqrt{a} \times \sqrt{a} = a$ .

Does this rule work for any number under the square root?

Absolutely! $\sqrt{5} \times \sqrt{5} = 5$ , $\sqrt{12} \times \sqrt{12} = 12$ , and so on. When you multiply a square root by itself, you always get the original number under the radical.

How do I remember this rule?

Think of square roots as "asking a question": $\sqrt{8}$ asks "what number squared gives 8?" When you multiply $\sqrt{8} \times \sqrt{8}$ , you're answering that question - the answer is 8!

What if I need to multiply different square roots like √8 × √2?

Then you do use the property $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$ . So $\sqrt{8} \times \sqrt{2} = \sqrt{16} = 4$ . The "shortcut rule" only applies when multiplying identical square roots.

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Multiply Square Roots: Solving √8 × √8

Square Root Multiplication with Same Radicands

❤️ 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 √8 × √8 equal 8 and not √64?

Can I use the property √a × √b = √(a×b) here?

Does this rule work for any number under the square root?

How do I remember this rule?

What if I need to multiply different square roots like √8 × √2?

🌟 Unlock Your Math Potential

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