Simplify the Square Root Ratio: √a:√b - Step-by-Step Solution

Square Root Quotients with Ratio Notation

Choose the expression that is equal to the following:

a:b \sqrt{a}:\sqrt{b}

❤️ 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 find expressions that are the same.
00:13 First, write the division as a fraction.
00:16 Divide the root of the numerator called M, by the root of the denominator called N.
00:22 This gives us the root of the fraction, which is M divided by N.
00:27 Now, apply this formula to our exercise.
00:32 And here we have the solution! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Choose the expression that is equal to the following:

a:b \sqrt{a}:\sqrt{b}

2

Step-by-step solution

To solve the problem, we will apply the rules of roots, specifically the Square Root Quotient Property:

  • Step 1: The given expression is a:b\sqrt{a}:\sqrt{b}, which represents the division of the square roots.
  • Step 2: Apply the square root quotient property: ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}.
  • Step 3: In terms of ratio notation, a:b\sqrt{a}:\sqrt{b} simplifies to a:b\sqrt{a:b}.

Therefore, the expression a:b\sqrt{a}:\sqrt{b} is equivalent to a:b\sqrt{a:b}, which is represented by choice 1.

3

Final Answer

a:b \sqrt{a:b}

Key Points to Remember

Essential concepts to master this topic
  • Quotient Rule: Division of square roots equals square root of quotient
  • Technique: Transform a:b \sqrt{a}:\sqrt{b} into a:b \sqrt{a:b}
  • Check: Verify using ab=ab \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} property ✓

Common Mistakes

Avoid these frequent errors
  • Treating ratio notation like subtraction or multiplication
    Don't change a:b \sqrt{a}:\sqrt{b} to a−b \sqrt{a}-\sqrt{b} or a⋅b \sqrt{a \cdot b} = completely wrong operations! The colon represents division, not subtraction or multiplication. Always recognize that ratio notation means division and apply the quotient property.

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

What does the colon symbol mean in math?

+

The colon (:) represents ratio or division. So a:b \sqrt{a}:\sqrt{b} means ab \frac{\sqrt{a}}{\sqrt{b}} , just like saying "a divided by b".

Why can I move the square root outside the ratio?

+

This follows the Square Root Quotient Property: ab=ab \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} . The square root of a quotient equals the quotient of square roots!

Is a:b \sqrt{a:b} the same as a:b \sqrt{a}:\sqrt{b} ?

+

Yes! They are equivalent expressions. The quotient property lets you move the square root symbol in or out of the ratio operation.

What if a or b are negative numbers?

+

Be careful! For real numbers, you can only take square roots of non-negative values. Both a and b must be positive for this property to work with real square roots.

Does this work with other roots like cube roots?

+

Absolutely! The quotient property works for any root: abn=anbn \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} for any positive integer n.

🌟 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

More Questions

Click on any question to see the complete solution with step-by-step explanations

Square Root Quotient Property

Solve: Multiplying Square Roots (√70 × √10)/√7 Simplification
Click to solve
Solve Fourth Root Division: Simplifying (Fourth Root of 128)/(Fourth Root of 8)
Click to solve