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Chapter 10: Problem 72

Approximate. Round to the nearest thousandth. $$ \sqrt[6]{13} $$

Short Answer

Expert verified
1.516

Step by step solution

01

Identify the root to be calculated

Recognize that the exercise requires finding the sixth root of 13, denoted as \(\root[6]{13}\).
02

Set up the calculation

To approximate the sixth root of 13, input or calculate \(\root[6]{13}\) using a calculator or computational tool capable of finding roots with high precision.
03

Perform the calculation

Calculate \(\root[6]{13}\) using the tool. Ensure the value is rounded to the nearest thousandth.
04

Round the result

Take the computed value (which should be approximately 1.5162) and round it to the nearest thousandth. In this case, 1.5162 rounded to the nearest thousandth is 1.516.

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Key Concepts

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

Sixth Root
When talking about the sixth root, we are dealing with finding a number that, when multiplied by itself six times, gives us the original number. In this case, we want to figure out \(\root[6]{13}\). Think of it as the reverse operation of raising a number to the sixth power. For example, if \(x\) is the sixth root of 13, then \(x^6 = 13\). We use this concept often in mathematics to simplify and solve equations that involve higher powers.
Rounding
Rounding involves adjusting a number to a specified level of precision. In this exercise, we need to round our result to the nearest thousandth. Let's break that down:
  • Nearest: This means we are selecting the closest value.
  • Thousandth: This refers to the third decimal place, or the value after three digits following the decimal point.
For instance, if we calculate the sixth root of 13 and get around 1.5162, we need to round this number to three decimal places. If the digit in the fourth decimal place (2 in this case) is 5 or higher, we would round the third decimal place up by one. Since it is below 5, we would keep the 1.516 as it is.
High Precision Calculation
High precision calculation means that our computations must be as accurate as possible. This is particularly important in scientific and engineering contexts, where even minor errors can lead to significant issues. In our exercise, we need a precise method to find \(\root[6]{13}\). Most calculators and computational tools can perform this calculation, ensuring a high precision result.
Here鈥檚 how to approach it:
  • First, understand the mathematical operation you need鈥攆inding the sixth root here
  • Next, use a tool that supports high precision calculations
  • Finally, ensure your result is accurate by verifying and rounding it appropriately
By following these steps, we can accurately approximate the value of \(\root[6]{13}\) and round it correctly to the nearest thousandth.

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