Solve √(36/144) × √√16: Multiple Square Root Operations

Q: How do I handle the double square root in $ \sqrt{\sqrt{16}} $ ?

Work from the inside out! First find $ \sqrt{16} = 4 $, then take the square root of that result: $ \sqrt{4} = 2 $. Never try to skip steps with nested radicals.

Q: Can I simplify the fraction before taking the square root?

Yes, absolutely! Simplifying $ \frac{36}{144} = \frac{1}{4} $ first makes $ \sqrt{\frac{1}{4}} = \frac{1}{2} $ much easier to see. Always look for ways to simplify fractions first.

Q: Why does $ \frac{1}{2} \times 2 = 1 $ ?

When you multiply a number by its reciprocal (flip), you always get 1! Since $ \frac{1}{2} $ and 2 are reciprocals, their product equals 1.

Q: What if I calculated $ \sqrt{\frac{36}{144}} $ as a decimal first?

You could find $ \sqrt{0.25} = 0.5 $, but working with exact fractions like $ \frac{1}{2} $ is usually cleaner and avoids rounding errors in your final answer.

Q: How can I check my work on problems like this?

Work backwards! Start with your answer of 1, and verify: $ \frac{1}{2} \times 2 = 1 $ ✓, $ \sqrt{4} = 2 $ ✓, and $ \sqrt{\frac{1}{4}} = \frac{1}{2} $ ✓.

Nested Square Roots with Fraction Operations

Solve the following exercise:

$\sqrt{\frac{36}{144}}\cdot\sqrt{\sqrt{16}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:03 The root of a fraction (A divided by B)

00:06 Equals the root of the numerator (A) divided by the root of the denominator (B)

00:10 We'll apply this formula to our exercise

00:20 Break down 36 to 6 squared

00:23 Break down 144 to 12 squared

00:27 Break down 16 to 4 squared

00:31 The root cancels the square

00:44 Break down 12 into factors of 6 and 2

00:48 Break down 4 into 2 squared

00:52 Simplify wherever possible

00:55 The root cancels the square

01:00 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt{\frac{36}{144}}\cdot\sqrt{\sqrt{16}}=$

Step-by-step solution

To solve the expression $\sqrt{\frac{36}{144}} \cdot \sqrt{\sqrt{16}}$ , follow these steps:

Simplify $\sqrt{\frac{36}{144}}$ :
- Evaluate the fraction: $\frac{36}{144} = \frac{1}{4}$ .
- Take the square root: $\sqrt{\frac{1}{4}} = \frac{1}{2}$ because $\sqrt{1} = 1$ and $\sqrt{4} = 2$ .
Simplify $\sqrt{\sqrt{16}}$ :
- First evaluate the inner square root: $\sqrt{16} = 4$ since $4^2 = 16$ .
- Then take the square root of the result: $\sqrt{4} = 2$ since $2^2 = 4$ .
Multiply the results from both parts:
- Multiply the simplified results: $\frac{1}{2} \cdot 2 = 1$ .

Therefore, the solution to the expression is $1$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Order: Simplify nested radicals from innermost to outermost operation
Technique: Evaluate $\sqrt{\sqrt{16}} = \sqrt{4} = 2$ step by step
Check: Final calculation $\frac{1}{2} \cdot 2 = 1$ verifies our answer ✓

Common Mistakes

Avoid these frequent errors

Attempting to combine radicals before simplifying each part separately
Don't try to combine $\sqrt{\frac{36}{144}} \cdot \sqrt{\sqrt{16}}$ into one radical = confusing mess! This creates complex nested expressions that are nearly impossible to solve. Always simplify each radical expression completely before multiplying the results together.

Practice Quiz

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FAQ

Everything you need to know about this question

How do I handle the double square root in $\sqrt{\sqrt{16}}$ ?

Work from the inside out! First find $\sqrt{16} = 4$ , then take the square root of that result: $\sqrt{4} = 2$ . Never try to skip steps with nested radicals.

Can I simplify the fraction before taking the square root?

Yes, absolutely! Simplifying $\frac{36}{144} = \frac{1}{4}$ first makes $\sqrt{\frac{1}{4}} = \frac{1}{2}$ much easier to see. Always look for ways to simplify fractions first.

Why does $\frac{1}{2} \times 2 = 1$ ?

When you multiply a number by its reciprocal (flip), you always get 1! Since $\frac{1}{2}$ and 2 are reciprocals, their product equals 1.

What if I calculated $\sqrt{\frac{36}{144}}$ as a decimal first?

You could find $\sqrt{0.25} = 0.5$ , but working with exact fractions like $\frac{1}{2}$ is usually cleaner and avoids rounding errors in your final answer.

How can I check my work on problems like this?

Work backwards! Start with your answer of 1, and verify: $\frac{1}{2} \times 2 = 1$ ✓, $\sqrt{4} = 2$ ✓, and $\sqrt{\frac{1}{4}} = \frac{1}{2}$ ✓.

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Solve √(36/144) × √√16: Multiple Square Root Operations

Nested Square Roots with Fraction Operations

❤️ 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

How do I handle the double square root in $\sqrt{\sqrt{16}}$ ?

Can I simplify the fraction before taking the square root?

Why does $\frac{1}{2} \times 2 = 1$ ?

What if I calculated $\sqrt{\frac{36}{144}}$ as a decimal first?

How can I check my work on problems like this?

🌟 Unlock Your Math Potential

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Solve √(36/144) × √√16: Multiple Square Root Operations

Nested Square Roots with Fraction Operations

❤️ 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

How do I handle the double square root in 16 \sqrt{\sqrt{16}} 16​​?

Can I simplify the fraction before taking the square root?

Why does 12×2=1 \frac{1}{2} \times 2 = 1 21​×2=1?

What if I calculated 36144 \sqrt{\frac{36}{144}} 14436​​ as a decimal first?

How can I check my work on problems like this?

🌟 Unlock Your Math Potential

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Solve the Expression: Cube Root of Square Root of 64 Times Square Root of 64

How do I handle the double square root in $\sqrt{\sqrt{16}}$ ?

Why does $\frac{1}{2} \times 2 = 1$ ?

What if I calculated $\sqrt{\frac{36}{144}}$ as a decimal first?