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

Q: Why do I work from inside to outside with nested radicals?

Working from inside out follows the order of operations! Just like parentheses, you must simplify the innermost expression first: $ \sqrt[5]{x^{30}} = x^6 $, then apply the outer radical.

Q: How do I remember the radical-to-exponent rule?

Think of it as "power over root": $ \sqrt[n]{a^m} = a^{m/n} $. The original power becomes the numerator, and the root index becomes the denominator.

Q: Why is the answer $ x^2 = 12 $ instead of $ x = 12 $ ?

After simplifying both sides, we get $ 12 = x^2 $. We're asked to solve for the relationship, not find the exact value of x. The answer $ x^2 = 12 $ is the complete solution.

Q: What does $ x^{10 \cdot 3} $ mean in the original equation?

This means $ x^{30} $ because $ 10 \times 3 = 30 $. The multiplication is done first in the exponent, giving us $ \sqrt[3]{\sqrt[5]{x^{30}}} $.

Nested Radicals with Exponential Simplification

Solve the following exercise:

$\sqrt{144}=\sqrt[3]{\sqrt[5]{x^{10\cdot3}}}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the value X

00:06 Solve the power multiplication

00:16 Multiply the order of the first root by the order of the second root

00:22 Apply the order that we obtained as a root to our number

00:27 Apply this formula to our exercise

00:32 When we have a root of order (C) on a number (A) to the power of (B)

00:39 The result equals number (A) to the power of (B divided by C)

00:43 Apply this formula to our exercise

00:52 Break down 144 to 12 squared

00:57 The root cancels out the square

01:02 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:

$\sqrt{144}=\sqrt[3]{\sqrt[5]{x^{10\cdot3}}}$

Step-by-step solution

To solve this equation, we will follow these steps:

Step 1: Simplify the left side of the equation $\sqrt{144}$ .
Step 2: Simplify the right side of the equation $\sqrt[3]{\sqrt[5]{x^{30}}}$ .
Step 3: Equate the simplified expressions and solve for $x$ .

Let us go through these steps:

Step 1: Simplify the left side:

The left side of the equation is $\sqrt{144}$ , which simplifies to $12$ , because $\sqrt{144} = 12$ .

Step 2: Simplify the right side:

The expression on the right is $\sqrt[3]{\sqrt[5]{x^{30}}}$ . Let's simplify it step by step:

First, simplify $\sqrt[5]{x^{30}}$ :
- Using the rule $\sqrt[n]{a^m} = a^{m/n}$ , we have $\sqrt[5]{x^{30}} = x^{30/5} = x^6$ .
Next, simplify $\sqrt[3]{x^6}$ :
- Again using the same rule, $\sqrt[3]{x^6} = x^{6/3} = x^2$ .

Step 3: Equate and solve:

From the previous steps, we get:

$12 = x^2$

Therefore, the solution to the equation is:
$x^2 = 12$ .

Final Answer

$x^2=12$

Key Points to Remember

Essential concepts to master this topic

Radical Rule: $\sqrt[n]{a^m} = a^{m/n}$ converts roots to fractional exponents
Technique: Work from inside out: $\sqrt[5]{x^{30}} = x^{30/5} = x^6$
Check: Verify $x^2 = 12$ by testing that $12 = \sqrt[3]{\sqrt[5]{(\sqrt{12})^{30}}}$ ✓

Common Mistakes

Avoid these frequent errors

Working from outside to inside with nested radicals
Don't start with the cube root first = $(x^{30})^{1/3} = x^{10}$ , then try the fifth root! This creates more complex expressions. Always work from the innermost radical outward: $\sqrt[5]{x^{30}}$ first, then $\sqrt[3]{}$ .

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 work from inside to outside with nested radicals?

Working from inside out follows the order of operations! Just like parentheses, you must simplify the innermost expression first: $\sqrt[5]{x^{30}} = x^6$ , then apply the outer radical.

How do I remember the radical-to-exponent rule?

Think of it as "power over root": $\sqrt[n]{a^m} = a^{m/n}$ . The original power becomes the numerator, and the root index becomes the denominator.

Why is the answer $x^2 = 12$ instead of $x = 12$ ?

After simplifying both sides, we get $12 = x^2$ . We're asked to solve for the relationship, not find the exact value of x. The answer $x^2 = 12$ is the complete solution.

What does $x^{10 \cdot 3}$ mean in the original equation?

This means $x^{30}$ because $10 \times 3 = 30$ . The multiplication is done first in the exponent, giving us $\sqrt[3]{\sqrt[5]{x^{30}}}$ .

Can I solve this by raising both sides to different powers?

Yes, but it's more complex! You'd need to raise both sides to the 15th power (LCM of 3 and 5) to eliminate both radicals at once. The inside-out method is much cleaner.

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

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 work from inside to outside with nested radicals?

How do I remember the radical-to-exponent rule?

Why is the answer $x^2 = 12$ instead of $x = 12$ ?

What does $x^{10 \cdot 3}$ mean in the original equation?

Can I solve this by raising both sides to different powers?

🌟 Unlock Your Math Potential

More Questions

Rules of Roots Combined

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

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

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

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

Root of a Root

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

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

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

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 work from inside to outside with nested radicals?

How do I remember the radical-to-exponent rule?

Why is the answer x2=12 x^2 = 12 x2=12 instead of x=12 x = 12 x=12?

What does x10⋅3 x^{10 \cdot 3} x10⋅3 mean in the original equation?

Can I solve this by raising both sides to different powers?

🌟 Unlock Your Math Potential

More Questions

Rules of Roots Combined

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

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

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

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

Root of a Root

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

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

Why is the answer $x^2 = 12$ instead of $x = 12$ ?

What does $x^{10 \cdot 3}$ mean in the original equation?