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

Q: How do I simplify $ \sqrt{\sqrt{81}} $ step by step?

Work from the inside out: First find $ \sqrt{81} = 9 $, then $ \sqrt{9} = 3 $. You can also use exponents: $ \sqrt{\sqrt{81}} = 81^{\frac{1}{2} \cdot \frac{1}{2}} = 81^{\frac{1}{4}} = 3 $

Q: Can I use the rule $ \sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a} $ ?

Yes! This is a powerful shortcut: $ \sqrt[5]{\sqrt{x^{10}}} = \sqrt[5 \cdot 2]{x^{10}} = \sqrt[10]{x^{10}} = x $. Both methods give the same answer!

Q: What if x could be negative?

For real number solutions, we typically assume x ≥ 0 when dealing with even roots. If x were negative, $ x^{10} $ would be positive, but $ x^5 $ would be negative, complicating the radical simplification.

Q: How do I verify my answer x = 3?

Substitute back into both sides: Left side: $ \sqrt[5]{\sqrt{3^{10}}} = \sqrt[5]{\sqrt{59049}} = \sqrt[5]{243} = 3 $. Right side: $ \sqrt{\sqrt{81}} = \sqrt{9} = 3 $. Both equal 3! ✓

Nested Radical Simplification with Exponent Rules

Solve the following exercise:

$\sqrt[5]{\sqrt[]{x^{10}}}=\sqrt{\sqrt{81}}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's solve this problem, step by step.

00:12 A regular root, like a square root, is order 2. So, we break down 81 as 9 squared.

00:19 Remember, we multiply the orders of the roots together.

00:23 Now, use that new order as the root for our number.

00:27 Let's apply this idea to our problem.

00:32 See how the root cancels out the square nicely?

00:38 If we have a root of order C, on A to the power B, here's what we do.

00:44 The result is A to the power of B divided by C.

00:48 Let's try that with our problem again.

00:53 Break down 9 as 3 squared.

00:56 And once more, the root cancels the square.

01:00 And that's how we find our solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt[5]{\sqrt[]{x^{10}}}=\sqrt{\sqrt{81}}$

Step-by-step solution

To solve this problem, we'll begin by simplifying both sides of the equation:

Simplifying the left-hand side:

$\sqrt[5]{\sqrt{x^{10}}}$ can be rewritten using properties of exponents and roots. The inner square root is $(x^{10})^{1/2} = x^{10 \cdot \frac{1}{2}} = x^{5}$ .

Then, take the fifth root: $(x^{5})^{\frac{1}{5}} = x^{5 \cdot \frac{1}{5}} = x^{1} = x$ .
Thus, the left-hand side simplifies to $x$ .

Simplifying the right-hand side:

$\sqrt{\sqrt{81}}$ simplifies as follows: First, find $\sqrt{81} = 9$ .
Then, compute $\sqrt{9} = 3$ .

So, the equation reduces to $x = 3$ .

Therefore, the solution to the problem is $\boxed{x = 3}$ .

Final Answer

$x=3$

Key Points to Remember

Essential concepts to master this topic

Exponent Rule: Convert nested radicals to fractional exponents for easier manipulation
Technique: $\sqrt[5]{\sqrt{x^{10}}} = (x^{10})^{\frac{1}{2} \cdot \frac{1}{5}} = x^1 = x$
Check: Substitute x = 3: $\sqrt[5]{\sqrt{3^{10}}} = \sqrt[5]{\sqrt{59049}} = \sqrt[5]{243} = 3$ ✓

Common Mistakes

Avoid these frequent errors

Working from outside to inside with nested radicals
Don't start with the fifth root first = $\sqrt[5]{\sqrt{x^{10}}} ≠ (\sqrt[5]{\sqrt{}})x^{10}$ ! This breaks the order of operations and creates impossible expressions. Always work from the innermost radical outward: first simplify $\sqrt{x^{10}} = x^5$ , then take the fifth root.

Practice Quiz

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Choose the largest value

FAQ

Everything you need to know about this question

Why can I write $\sqrt{x^{10}} = x^5$ directly?

Since we're dealing with even exponents under square roots, we assume x ≥ 0 for real solutions. When x is positive, $\sqrt{x^{10}} = |x^5| = x^5$ because $x^5$ is positive.

How do I simplify $\sqrt{\sqrt{81}}$ step by step?

Work from the inside out: First find $\sqrt{81} = 9$ , then $\sqrt{9} = 3$ . You can also use exponents: $\sqrt{\sqrt{81}} = 81^{\frac{1}{2} \cdot \frac{1}{2}} = 81^{\frac{1}{4}} = 3$

Can I use the rule $\sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a}$ ?

Yes! This is a powerful shortcut: $\sqrt[5]{\sqrt{x^{10}}} = \sqrt[5 \cdot 2]{x^{10}} = \sqrt[10]{x^{10}} = x$ . Both methods give the same answer!

What if x could be negative?

For real number solutions, we typically assume x ≥ 0 when dealing with even roots. If x were negative, $x^{10}$ would be positive, but $x^5$ would be negative, complicating the radical simplification.

How do I verify my answer x = 3?

Substitute back into both sides: Left side: $\sqrt[5]{\sqrt{3^{10}}} = \sqrt[5]{\sqrt{59049}} = \sqrt[5]{243} = 3$ . Right side: $\sqrt{\sqrt{81}} = \sqrt{9} = 3$ . Both equal 3! ✓

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

Nested Radical Simplification with Exponent Rules

❤️ 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 can I write $\sqrt{x^{10}} = x^5$ directly?

How do I simplify $\sqrt{\sqrt{81}}$ step by step?

Can I use the rule $\sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a}$ ?

What if x could be negative?

How do I verify my answer x = 3?

🌟 Unlock Your Math Potential

More Questions

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Root of a Root

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

Nested Radical Simplification with Exponent Rules

❤️ 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 can I write x10=x5 \sqrt{x^{10}} = x^5 x10​=x5 directly?

How do I simplify 81 \sqrt{\sqrt{81}} 81​​ step by step?

Can I use the rule amn=anm \sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a} nma​​=nma​?

What if x could be negative?

How do I verify my answer x = 3?

🌟 Unlock Your Math Potential

More Questions

Rules of Roots Combined

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

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

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

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

Root of a Root

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

Solve the Square Root Equation: √98/√x = 7

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

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

Why can I write $\sqrt{x^{10}} = x^5$ directly?

How do I simplify $\sqrt{\sqrt{81}}$ step by step?

Can I use the rule $\sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a}$ ?