Solve Nested Square Roots: Simplifying √(√(81x⁴))

Q: What if x is negative?

For square roots to be real, we assume x ≥ 0. If x could be negative, we'd need absolute value notation: $ \sqrt{x^2} = |x| $.

Q: How do I know 81 = 9²?

Practice your perfect squares! Common ones: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100.

Q: Why does √(x⁴) = x²?

Because $ x^4 = (x^2)^2 $, so $ \sqrt{x^4} = \sqrt{(x^2)^2} = x^2 $. The square root 'cancels' with the square, leaving just x².

Q: Is there a shortcut for nested square roots?

Yes! $ \sqrt{\sqrt{a}} = a^{1/4} $ (fourth root). But showing the step-by-step method demonstrates your understanding and helps catch mistakes.

Nested Square Roots with Perfect Squares

Complete the following exercise:

$\sqrt{\sqrt{81\cdot x^4}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's solve this math problem together.

00:12 A regular root is usually of order two, like a square root.

00:19 When you have a number A, raised to the power of B, inside a root of order C.

00:26 The result is A, raised to the power of B divided by C.

00:32 Let's use this formula! We'll calculate by multiplying the orders.

00:40 For a root of a product like A times B.

00:45 We can split it into separate roots for each number.

00:48 Let's use this idea in our exercise to simplify the root.

00:53 We break down eighty-one into three raised to the power of four.

00:59 For a number like four, to the power of four, in a root of order C.

01:05 You get three, to the power of four divided by four.

01:09 Let's calculate these power quotients for our exercise.

01:13 And that's how we find the solution! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the following exercise:

$\sqrt{\sqrt{81\cdot x^4}}=$

Step-by-step solution

To solve the problem $\sqrt{\sqrt{81 \cdot x^4}}$ , we need to simplify this expression using properties of exponents and square roots.

Step 1: Simplify the inner square root
The expression inside the first square root is $81 \cdot x^4$ . We can rewrite this using exponents:
$81 = 9^2$ and $x^4 = (x^2)^2$ . Thus, $81 \cdot x^4 = (9x^2)^2$ .
Step 2: Apply the inner square root
Taking the square root of $(9x^2)^2$ gives us:
$\sqrt{(9x^2)^2} = 9x^2$ , because $\sqrt{a^2} = a$ where $a$ is a non-negative real number.
Step 3: Simplify the outer square root
Now, we take the square root of the result from the inner root:
$\sqrt{9x^2} = \sqrt{9} \cdot \sqrt{x^2} = 3 \cdot x = 3x$ , since $\sqrt{x^2} = x$ given $x$ is non-negative.

Therefore, the solution to the problem is $3x$ .

Final Answer

$3x$

Key Points to Remember

Essential concepts to master this topic

Rule: $\sqrt{\sqrt{a}} = a^{1/4}$ for nested radicals
Technique: Simplify inner root first: $\sqrt{81x^4} = 9x^2$ , then outer
Check: Verify by working backwards: $(3x)^4 = 81x^4$ ✓

Common Mistakes

Avoid these frequent errors

Taking the fourth root of 81 directly
Don't calculate $\sqrt{\sqrt{81x^4}} = (81x^4)^{1/4} = 3x$ in one step without showing work! This skips the conceptual understanding of nested radicals. Always simplify the inner square root first, then apply the outer root step by step.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt[5]{\sqrt[3]{5}}=$

FAQ

Everything you need to know about this question

Why can't I just take the fourth root of everything at once?

While mathematically equivalent, breaking it into steps helps you understand the concept better! Working step-by-step prevents errors and shows you recognize that $\sqrt{\sqrt{a}} = a^{1/4}$ .

What if x is negative?

For square roots to be real, we assume x ≥ 0. If x could be negative, we'd need absolute value notation: $\sqrt{x^2} = |x|$ .

How do I know 81 = 9²?

Practice your perfect squares! Common ones: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100.

Why does √(x⁴) = x²?

Because $x^4 = (x^2)^2$ , so $\sqrt{x^4} = \sqrt{(x^2)^2} = x^2$ . The square root 'cancels' with the square, leaving just x².

Is there a shortcut for nested square roots?

Yes! $\sqrt{\sqrt{a}} = a^{1/4}$ (fourth root). But showing the step-by-step method demonstrates your understanding and helps catch mistakes.

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