Solve Square Root Multiplication: √7 × √7 Simplification

Q: Why does √7 × √7 equal 7 and not √14?

When you multiply identical square roots, you're essentially squaring the square root: $ (\sqrt{7})^2 = 7 $. The √14 would only happen if you had √7 × √2, not √7 × √7.

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

Yes! Using this rule: $ \sqrt{7} \times \sqrt{7} = \sqrt{7 \times 7} = \sqrt{49} = 7 $. Both methods give the same answer, but recognizing identical square roots makes it faster.

Q: What if the problem was √7 × √8 instead?

Then you'd get $ \sqrt{7 \times 8} = \sqrt{56} $, which cannot be simplified to a whole number. The key difference is that √7 × √7 involves identical square roots.

Q: Is 7^(1/2) wrong as an answer?

It's mathematically correct but not simplified. The question asks you to solve the multiplication, so you should simplify $ 7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}} $ all the way to 7.

Square Root Multiplication with Same Radicands

Solve the following exercise:

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

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's solve this math problem together.

00:14 Remember, taking a root is like raising to an inverse power.

00:19 Now, let's apply this idea to our exercise.

00:23 So, a regular root is like a square root, or root of 2.

00:29 We'll change each number to its half power, or inverse power.

00:34 When you multiply powers with the same base, listen carefully.

00:38 The exponent of the result equals the sum of the exponents.

00:43 Let's use this rule and add up the powers in our exercise.

00:48 Great job! This gives us the solution. Well done!

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 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':

$\sqrt{7}\cdot\sqrt{7}= \\ \downarrow\\ 7^{\frac{1}{2}}\cdot7^{\frac{1}{2}}=$ 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:

$7^{\frac{1}{2}}\cdot7^{\frac{1}{2}}= \\ (7^{\frac{1}{2}})^2=\\ 7^{\frac{1}{2}\cdot2}=\\ 7^1=\\ \boxed{7}$ 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 $\sqrt{7} \cdot \sqrt{7}$ to $(\sqrt{7})^2 = 7$
Check: Verify that $7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}} = 7^{\frac{1}{2} + \frac{1}{2}} = 7^1 = 7$ ✓

Common Mistakes

Avoid these frequent errors

Leaving the answer as a square root expression
Don't write √7 × √7 = √14 or keep it as √7! This ignores the fundamental property that multiplying identical square roots eliminates the radical. Always recognize that √a × √a = a, giving you the number inside the radical.

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 √14?

When you multiply identical square roots, you're essentially squaring the square root: $(\sqrt{7})^2 = 7$ . The √14 would only happen if you had √7 × √2, not √7 × √7.

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

Yes! Using this rule: $\sqrt{7} \times \sqrt{7} = \sqrt{7 \times 7} = \sqrt{49} = 7$ . Both methods give the same answer, but recognizing identical square roots makes it faster.

How do I know when to convert to exponent form?

Converting to exponents helps when the multiplication isn't obvious. $\sqrt{7} = 7^{\frac{1}{2}}$ , so $7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}} = 7^{\frac{1}{2} + \frac{1}{2}} = 7^1 = 7$ .

What if the problem was √7 × √8 instead?

Then you'd get $\sqrt{7 \times 8} = \sqrt{56}$ , which cannot be simplified to a whole number. The key difference is that √7 × √7 involves identical square roots.

Is 7^(1/2) wrong as an answer?

It's mathematically correct but not simplified. The question asks you to solve the multiplication, so you should simplify $7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}$ all the way to 7.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Rules of Roots questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Solve Square Root Multiplication: √7 × √7 Simplification

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 √7 × √7 equal 7 and not √14?

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

How do I know when to convert to exponent form?

What if the problem was √7 × √8 instead?

Is 7^(1/2) wrong as an answer?

🌟 Unlock Your Math Potential

More Questions

Rules of Roots Combined

Solve: Multiplication of Sixth Root and Cube Root of 12

Solve Fourth and Sixth Roots: Multiplying √⁴3 × √⁶3

Product Property of Square Roots

Multiply Cube Roots: Solving ∛5 × ∛5

Multiply Fourth and Seventh Roots: Solving ⁴√8 · ⁷√8

Solving the Square Root Equation: Find (x, y) for √x - √y = √(√61 - 6) and xy = 9

Rectangle Side Ratio Problem: Finding Length m Using sqrt(x/2)