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Chapter 9: Problem 6

Estimate each square root to the nearest integer. Do not use a calculator. $$\sqrt{66}$$

Short Answer

Expert verified
The square root of 66, rounded to the nearest integer, is 8.

Step by step solution

01

Identify Perfect Squares Around the Number

We need to find perfect squares that 66 is between. The perfect squares around 66 are 64 and 81 because 64 is the square of 8 ( 8^2 = 64 ) and 81 is the square of 9 ( 9^2 = 81 ). So, 64 < 66 < 81 .
02

Determine Closest Perfect Square

Next, we estimate which of these perfect squares (64 or 81) is closer to 66. Since 66 - 64 = 2 and 81 - 66 = 15 , we see that 66 is closer to 64. Therefore, the square root of 66 is closer to 8.
03

Check Integer Midpoint Relevance

Let's check the midpoint, which is 8.5, to verify if either side is closer. If the square root was more accurately between integers at the midpoint, it reflects that the closer integer might slightly shift. However, since 66 is quite close to 64, it confirms that 8 is the nearest integer estimation.

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Key Concepts

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

Perfect Squares
Perfect squares are the result of multiplying an integer by itself. For instance, since 3 multiplied by 3 gives us 9, 9 is a perfect square. This concept is crucial because it helps us judge how close a number is to the square root we seek to estimate.
Perfect squares provide benchmarks or reference points. By identifying them, we can determine the range within which our target number lies.
  • Example: 4 is a perfect square because 2 times 2 equals 4.
  • Another example: 16 is a perfect square from 4 times 4.
When estimating square roots, like \(\sqrt{66}\), knowing the perfect squares around 66 allows us to figure out it lies between \(8^2 = 64\) and \(9^2 = 81\). By using these perfect squares, we can estimate that the square root of 66 falls between 8 and 9.
Integer Approximation
Integer approximation involves rounding a number to the nearest integer to simplify calculations or predictions. While estimating square roots, your goal is to find the nearest integer that the square root approaches.
To approximate \(\sqrt{66}\), consider the perfect squares around 66, which are 64 (from 8) and 81 (from 9). Since 66 is nearer to 64 than to 81, we determine that \(\sqrt{66}\) most closely approximates to 8.
Use integer approximation when calculations need simplification, especially when a precise calculator isn't handy. This keeps math approachable:
  • Round off calculations for quicker estimation.
  • Avoid over-complicating problems by using close estimates.
Mathematical Estimation
Mathematical estimation is an essential skill that helps us make quick decisions without precise tools. It involves using nearby known values to approximate a less simple figure.
In the context of square roots, follow these steps:
  • Identify perfect squares on either side of your target, as seen with \(\sqrt{66}\) nestled between 64 and 81.
  • Compare the target number (66) with these squares to see which is closer. Here, 66 is closer to 64.
  • Decide if it's closer to the lower or upper perfect square, allowing you to guess the integer value more easily.
Estimation is an everyday tool, empowering you to make judgments based on nearby 'clue' numbers, rather than exact figures.

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Most popular questions from this chapter

Name all of the sets of numbers to which each real number belongs. Let \(\mathbf{N}=\) natural numbers, \(\mathbf{W}=\) whole numbers, \(\mathbf{Z}=\) integers, \(\mathbf{Q}=\) rational numbers, and I = irrational numbers. $$-\frac{56}{8}$$

Solve each equation. Round to the nearest tenth, if necessary. $$3 m^{2}=222$$

Use a calculator to find each square root to the nearest tenth. \(-\sqrt{32}\)

Estimate each square root to the nearest whole number. Do not use a calculator. $$\sqrt{8.67}$$

Writing in Math Explain the relationship between the area of a square and the length of its sides. Give an example of a square whose side length is irrational and an example of a square whose side length is rational.
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