Multiply Square Roots: Solving √7 × √7

Q: Why does √7 × √7 equal 7 and not 49?

Think of it this way: $ \sqrt{7} $ asks "what number times itself equals 7?" So when you multiply $ \sqrt{7} \times \sqrt{7} $, you get 7, not 49!

Q: Can I use this rule with different square roots?

This specific rule only works with identical square roots. For different ones like $ \sqrt{3} \times \sqrt{5} $, you get $ \sqrt{15} $ using $ \sqrt{a} \times \sqrt{b} = \sqrt{ab} $.

Q: How do I remember this rule?

Remember: √a × √a = a. The square root "undoes" the squaring! It's like asking "what squared gives me 7?" and then actually doing that multiplication.

Q: What if I convert to decimals first?

You could calculate $ \sqrt{7} \approx 2.646 $ and multiply, but you'd get approximately 7. The exact answer is exactly 7 using the square root rule!

Q: Does this work with cube roots too?

Yes! The pattern continues: $ \sqrt[3]{a} \times \sqrt[3]{a} \times \sqrt[3]{a} = a $. You need three identical cube roots to get back to the original number.

Square Root Multiplication with Identical Radicals

Solve the following exercise:

$\sqrt{7}\cdot\sqrt{7}=$

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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 When we multiply the square root of A with the square root of B.

00:15 It's the same as the square root of A times B.

00:19 Now, let's apply this formula to our example.

00:23 Find the square root of forty-nine.

00:25 Great job! That's how we solve this type of problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt{7}\cdot\sqrt{7}=$

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 multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

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

$\sqrt{7}\cdot\sqrt{7}= \\ \downarrow\\ 7^{\frac{1}{2}}\cdot7^{\frac{1}{2}}=$ We'll continue, since we are multiplying two terms with identical bases - we'll use the law of exponents mentioned in b:

$7^{\frac{1}{2}}\cdot7^{\frac{1}{2}}= \\ 7^{\frac{1}{2}+\frac{1}{2}}=\\ 7^1=\\ \boxed{7}$ Therefore, the correct answer is answer a.

Final Answer

$7$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying identical square roots, the result equals the radicand
Technique: Convert to exponents: $\sqrt{7} \cdot \sqrt{7} = 7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}} = 7^1 = 7$
Check: Verify that 7 × 7 = 49, so $\sqrt{49} = 7$ ✓

Common Mistakes

Avoid these frequent errors

Adding the radicands instead of using multiplication rules
Don't think $\sqrt{7} \cdot \sqrt{7} = \sqrt{7+7} = \sqrt{14}$ = wrong answer! Square root multiplication doesn't work like addition. Always use the rule that $\sqrt{a} \cdot \sqrt{a} = a$ .

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 √7 × √7 equal 7 and not 49?

Think of it this way: $\sqrt{7}$ asks "what number times itself equals 7?" So when you multiply $\sqrt{7} \times \sqrt{7}$ , you get 7, not 49!

Can I use this rule with different square roots?

This specific rule only works with identical square roots. For different ones like $\sqrt{3} \times \sqrt{5}$ , you get $\sqrt{15}$ using $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ .

How do I remember this rule?

Remember: √a × √a = a. The square root "undoes" the squaring! It's like asking "what squared gives me 7?" and then actually doing that multiplication.

What if I convert to decimals first?

You could calculate $\sqrt{7} \approx 2.646$ and multiply, but you'd get approximately 7. The exact answer is exactly 7 using the square root rule!

Does this work with cube roots too?

Yes! The pattern continues: $\sqrt[3]{a} \times \sqrt[3]{a} \times \sqrt[3]{a} = a$ . You need three identical cube roots to get back to the original number.

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Multiply Square Roots: Solving √7 × √7

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 √7 × √7 equal 7 and not 49?

Can I use this rule with different square roots?

How do I remember this rule?

What if I convert to decimals first?

Does this work with cube roots too?

🌟 Unlock Your Math Potential

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