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Chapter 0: Problem 110

In Exercises \(109-110\), evaluate each expression. $$\sqrt[3]{\sqrt{\sqrt{169}+\sqrt{9}}+\sqrt{\sqrt[3]{1000}}+\sqrt[3]{216}}$$

Short Answer

Expert verified
The evaluated expression gives \(\sqrt[3]{17}\).

Step by step solution

01

Identify and Simplify the Innermost Roots

First, simplify the innermost square roots and cube roots. \n For the square roots, \(\sqrt{169} = 13\) and \(\sqrt{9} = 3\). \n For the cube root, \(\sqrt[3]{1000} = 10\) and \(\sqrt[3]{216} = 6\).
02

Compute the Sums and the Subsequent Roots

Next, add up the square roots and cube roots obtained from step 1 and compute the subsequent roots. \nWhich gives us \(\sqrt[3]{\sqrt{13 + 3}} + 10 + 6 = \sqrt[3]{\sqrt{16}} + 10 + 6\), \n and \(\sqrt[3]{4} + 10 + 6 = 1 + 10 + 6 = 17\).
03

Evaluate the Remaining Cube Root

Finally, evaluate the cube root of the resultant sum which is \(\sqrt[3]{17}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Square Roots
Square roots help you find a number that, when multiplied by itself, gives you the original number. For example, \( \sqrt{169} = 13 \) because \(13 \times 13 = 169\). Similarly, \( \sqrt{9} = 3 \), as \(3 \times 3 = 9\).
It's important to:
  • Recognize perfect squares like 169 and 9 for smoother calculations.
  • Understand that the square root operation is about reversing multiplication.
  • Visualize it by asking: "What number squared gives me this value?"
Recognizing these patterns allows you to simplify expressions involving square roots more easily.
Unveiling Cube Roots
Cube roots are a bit different than square roots. Here, you're looking for a number that, when used three times in a multiplication, gives you the original number.
For instance:
  • \( \sqrt[3]{1000} = 10 \) because \(10 \times 10 \times 10 = 1000\).
  • \( \sqrt[3]{216} = 6 \), since \(6 \times 6 \times 6 = 216\).
Cube roots are useful in finding dimensions in geometry and understanding volumes. Simplifying cube roots requires you to know cubes of numbers (e.g., 1, 8, 27, etc.). Knowing these can help you deal with complex expressions more easily.
Simplifying Radicals and Expressions
Simplifying radicals can sometimes seem tricky, but with practice, it becomes straightforward.
Here's how to handle them efficiently:
  • Focus on simplifying innermost roots first. This helps in breaking down complex expressions step-by-step.
  • Combine like terms after simplifying individual roots. For the given expression, we began by simplifying \( \sqrt{13 + 3} \) to \( \sqrt{16} \), which equals 4.
  • Always remember to apply arithmetic operations in the right order after simplifications.
By addressing radicals and expressions in parts and simplifying methodically, evaluating them becomes a simpler task. This method also helps you avoid common mistakes by maintaining clarity throughout your calculations.

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