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Chapter 7: Problem 10

Simplify. $$ \sqrt[3]{54}+\sqrt[3]{16} $$

Short Answer

Expert verified
The simplified result of \( \sqrt[3]{54} + \sqrt[3]{16} \) is 6.3.

Step by step solution

01

Finding the Cube Root of 54

Start by calculating the cube root of 54. The cube root of a number \(x\) is the value that when cubed (i.e., raised to the power of 3) gives the original number \(x\). In this case, \( \sqrt[3]{54} = 3.78 \) (rounded to two decimal places)
02

Finding the Cube Root of 16

Next, calculate the cube root of 16. Similarly, we find that \( \sqrt[3]{16} = 2.52 \) (rounded to two decimal places)
03

Add the Cube Roots

Finally, add the cube roots together. The sum of 3.78 and 2.52 equals 6.3.

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Key Concepts

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

Simplification of Cube Roots
Simplifying cube roots means finding an equivalent number that when raised to the power of three gives the same original number. In our example, we had to simplify \( \sqrt[3]{54} \).
The goal here is to find a number which, when cubed, will closely approximate 54. Similarly, for \( \sqrt[3]{16} \), identify a value whose cube gives 16.
Simplification is crucial as it forms the basis for easier manipulation in equations or further arithmetic operations.
  • The cube root of 54 is approximately 3.78.
  • The cube root of 16 is approximately 2.52.
Understanding this step makes the process of addition or further calculations simpler and more manageable.
Addition of Cube Root Radicals
After finding the cube roots, the next step involves adding these radicals together. When handling radical expressions, like cube roots, addition isn't quite the same as with integers or decimals.
This is because each radical represents a unique value which directly depends on the root and the original number.
In our exercise, the radicals were calculated as \( \sqrt[3]{54} \approx 3.78 \) and \( \sqrt[3]{16} \approx 2.52 \).
To perform the addition:
  • Simply add the numerical values of the cube roots: 3.78 + 2.52.
  • This gives a total of 6.3.
By adding these radicals numerically, we've combined their unique contributions to get a simplified result.
Numerical Approximations for Cube Roots
When exact values for cube roots aren't available, numerical approximations can help. They provide a practical estimation that's close to the actual value.
For situations where a precise calculation isn't possible or necessary, rounding to a certain precision ensures easier readability and processing.
For example,
  • \( \sqrt[3]{54} \approx 3.78 \) indicates that the true cube root is just a bit more or less than this value.
  • \( \sqrt[3]{16} \approx 2.52 \) similarly offers a close estimate.
Approximations like these are particularly useful in complex mathematical problems where exact calculation is cumbersome or inapplicable. By using them, we simplify the process while maintaining accuracy for practical purposes.

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