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Chapter 8: Problem 7

Evaluate. If the number is irrational, round to the nearest hundredth. $$\sqrt{101}$$

Short Answer

Expert verified
10.05

Step by step solution

01

Recognize the Problem

Determine whether the number inside the square root, \(\root{101}\), is a perfect square. Since 101 is not a perfect square, the result will be an irrational number.
02

Use Calculator

Use a calculator to find the square root of 101. Input \(\root{101}\) into the calculator.
03

Find the Exact Value

The calculator will provide the value of \(\root{101}\), which is approximately equal to 10.04987562112089.
04

Round to the Nearest Hundredth

Round the obtained value to the nearest hundredth. 10.04987562112089 rounded to the nearest hundredth is 10.05.

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Key Concepts

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

irrational numbers
An irrational number is a number that cannot be expressed as a simple fraction 鈥� meaning it has a non-repeating, non-terminating decimal expansion. For example, the square root of 101 is irrational because its decimal form goes on forever without repeating. When you try to find the square root of a non-perfect square using a calculator, you'll notice the result is an unending decimal. Concepts like this are fundamental because they help bridge the gap between simple arithmetic and more complex mathematics.
  • Irrational numbers have endless non-repeating decimals.
  • They cannot be precisely expressed as fractions.
  • Common examples include \(\root{2}\) and \(\root{\root{101}}\).
perfect square
A perfect square is a number that is the square of an integer. In other words, when you multiply an integer by itself, you get a perfect square. Numbers like 1, 4, 9, 16, and 25 are all perfect squares. Recognizing perfect squares can simplify problems that involve square roots. For example, \(\root{100}\) is 10 because 100 is a perfect square (10 \times 10 = 100). However, since 101 is not a perfect square (there is no integer that multiplies by itself to give 101), its square root is not an integer and instead falls under irrational numbers.
  • A perfect square results from squaring an integer.
  • Examples include 1, 4, 9, 16, and so on.
  • They simplify calculations involving roots.
nearest hundredth
Rounding to the nearest hundredth means shortening a number to two decimal places. This is essential when dealing with irrational numbers because their decimal form is infinite. For instance, the square root of 101, approximated by a calculator, is 10.04987562112089. To round to the nearest hundredth, you look at the third decimal place:
If the third digit is 5 or higher, you round up the second decimal place by one. If it is 4 or lower, you keep the second decimal place as is. Here, the third digit in 10.04987562112089 is 9, so rounding up is required: 10.05.
  • Focus on the third decimal for rounding.
  • Round up if the third decimal is 5 or more.
  • Keep the second decimal the same if the third is less than 5.

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