Solve the Nested Radical Equation: √√81 = ∛√x^6

Q: Why do I start with the innermost radical first?

Working inside-out prevents confusion with exponent rules! Start with $ \sqrt{81} = 9 $, then $ \sqrt{9} = 3 $. This keeps each step clear and manageable.

Q: How do I know when $ \sqrt{x^6} = x^3 $ ?

Use the rule $ \sqrt{a} = a^{1/2} $! So $ \sqrt{x^6} = (x^6)^{1/2} = x^{6 \cdot 1/2} = x^3 $. Always multiply the exponents when raising a power to a power.

Q: What if I get a negative answer for x?

In this problem, we're dealing with even roots like square roots, which typically give positive results. Since $ \sqrt{\sqrt{81}} = 3 $ is positive, x should be positive too.

Q: Can I solve this without converting to exponential form?

Yes, but it's much harder! Converting radicals to exponential form like $ \sqrt[3]{\sqrt{x^6}} = (x^6)^{1/6} = x $ makes the algebra cleaner and less error-prone.

Q: How do I check my answer is correct?

Substitute your value back into both sides of the original equation! With x = 3: Left side = $ \sqrt{\sqrt{81}} = 3 $, Right side = $ \sqrt[3]{\sqrt{3^6}} = \sqrt[3]{\sqrt{729}} = \sqrt[3]{27} = 3 $. They match!

Nested Radicals with Exponential Simplification

Solve the following exercise:

$\sqrt{\sqrt{81}}=\sqrt[3]{\sqrt{x^6}}$

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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 First, break down 81 into 9 to the power of 2.

00:17 Remember, a regular root is like a root of order 2.

00:22 The square root cancels out the square. Isn't that neat?

00:29 Next, multiply the order of the first root by the order of the second root.

00:34 Now, apply this order as a root to our number.

00:38 Let's put this formula into practice with our exercise.

00:42 When you have a root of order C on number A to the power of B,

00:47 the result is A raised to the power of B divided by C.

00:52 Let's use this formula in our exercise again.

00:55 Break down 9 into 3 to the power of 2.

00:59 The square root cancels the square. See how it's done?

01:03 And that's how we find the solution! Great work!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt{\sqrt{81}}=\sqrt[3]{\sqrt{x^6}}$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Simplify the left side of the equation, $\sqrt{\sqrt{81}}$ .
Step 2: Simplify the right side of the equation, $\sqrt[3]{\sqrt{x^6}}$ .
Step 3: Equate the simplified expressions and solve for $x$ .

Now, let's simplify each side:

Step 1: Simplify $\sqrt{\sqrt{81}}$ .

First, evaluate $\sqrt{81}$ , which is $9$ , since $9^2 = 81$ .
Then, evaluate $\sqrt{9}$ , which is $3$ , since $3^2 = 9$ .
So, $\sqrt{\sqrt{81}} = 3$ .

Step 2: Simplify $\sqrt[3]{\sqrt{x^6}}$ .

Express $\sqrt{x^6}$ as $(x^6)^{1/2} = x^{6/2} = x^3$ .
Express $\sqrt[3]{x^3}$ as $(x^3)^{1/3} = x^{3/3} = x^1 = x$ .
So, $\sqrt[3]{\sqrt{x^6}} = x$ .

Step 3: Set the simplified expressions equal.

We have simplified both sides of the equation to get $3 = x$ .
Therefore, the solution to the problem is $x = 3$ .

Hence, the correct answer is $x = 3$ .

Therefore, the correct choice is:

Choice 2: $x = 3$ .

Final Answer

$x=3$

Key Points to Remember

Essential concepts to master this topic

Rule: Simplify nested radicals by working from inside out
Technique: Convert $\sqrt{x^6}$ to $x^3$ , then $\sqrt[3]{x^3} = x$
Check: Substitute x = 3: $\sqrt{\sqrt{81}} = 3$ and $\sqrt[3]{\sqrt{3^6}} = 3$ ✓

Common Mistakes

Avoid these frequent errors

Simplifying the entire expression at once without working step by step
Don't try to solve $\sqrt{\sqrt{81}} = \sqrt[3]{\sqrt{x^6}}$ all at once = confusion and errors! This leads to mixing up exponent rules and getting wrong answers. Always simplify each side separately first, then equate the results.

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 do I start with the innermost radical first?

Working inside-out prevents confusion with exponent rules! Start with $\sqrt{81} = 9$ , then $\sqrt{9} = 3$ . This keeps each step clear and manageable.

How do I know when $\sqrt{x^6} = x^3$ ?

Use the rule $\sqrt{a} = a^{1/2}$ ! So $\sqrt{x^6} = (x^6)^{1/2} = x^{6 \cdot 1/2} = x^3$ . Always multiply the exponents when raising a power to a power.

What if I get a negative answer for x?

In this problem, we're dealing with even roots like square roots, which typically give positive results. Since $\sqrt{\sqrt{81}} = 3$ is positive, x should be positive too.

Can I solve this without converting to exponential form?

Yes, but it's much harder! Converting radicals to exponential form like $\sqrt[3]{\sqrt{x^6}} = (x^6)^{1/6} = x$ makes the algebra cleaner and less error-prone.

How do I check my answer is correct?

Substitute your value back into both sides of the original equation! With x = 3: Left side = $\sqrt{\sqrt{81}} = 3$ , Right side = $\sqrt[3]{\sqrt{3^6}} = \sqrt[3]{\sqrt{729}} = \sqrt[3]{27} = 3$ . They match!

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Solve the Nested Radical Equation: √√81 = ∛√x^6

Nested Radicals with Exponential Simplification

❤️ 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 do I start with the innermost radical first?

How do I know when $\sqrt{x^6} = x^3$ ?

What if I get a negative answer for x?

Can I solve this without converting to exponential form?

How do I check my answer is correct?

🌟 Unlock Your Math Potential

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Solve the Nested Radical Equation: √√81 = ∛√x^6

Nested Radicals with Exponential Simplification

❤️ 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 do I start with the innermost radical first?

How do I know when x6=x3 \sqrt{x^6} = x^3 x6​=x3?

What if I get a negative answer for x?

Can I solve this without converting to exponential form?

How do I check my answer is correct?

🌟 Unlock Your Math Potential

More Questions

Root of a Root

Solve the Nested Root Equation: Fifth Root of Square Root of x^10 = √√81

Solve the Square Root Equation: x^6 = √16 · √25

Solve Nested Square Roots: √√(100/25) × √√25 Multiplication Problem

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

Rules of Roots Combined

Solve for X: Square Root of 144 Equals Nested Cube and Fifth Roots

Simplify the Expression: (√35 × √20) ÷ √7

Solve the Nested Radical Equation: √(16/∛64) = √(x²)

Simplify the Nested Radical: √(16/∛64) Step-by-Step Solution

How do I know when $\sqrt{x^6} = x^3$ ?