Solved Examples: Estimating Non-Perfect Square Roots (Grade 8 Math)

📚 Quick Study Guide

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• Perfect Squares: Understand perfect squares are numbers that result from squaring whole numbers (e.g., 1, 4, 9, 16...).
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• Estimating: To estimate $\sqrt{x}$ (where x is not a perfect square), find the two nearest perfect squares, $a^2$ and $b^2$, such that $a^2 < x < b^2$. Then, $a < \sqrt{x} < b$.
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• Refining the Estimate: Determine if $x$ is closer to $a^2$ or $b^2$. This helps in refining your estimate between $a$ and $b$.
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• Decimal Approximation: Use decimals to get closer to the actual value. Try numbers like $a.1, a.2, a.5$, and square them to see how close they are to $x$.
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• Averaging Method: You can also use the formula $\sqrt{x} ≈ \frac{x + a^2}{2a}$ where $a^2$ is the nearest perfect square.

Practice Quiz

1. Question 1: Which two perfect squares is $\sqrt{12}$ between?
   1. A) 2 and 3
   2. B) 3 and 4
   3. C) 9 and 16
   4. D) 10 and 14
2. Question 2: Estimate $\sqrt{20}$ to the nearest whole number.
   1. A) 3
   2. B) 4
   3. C) 5
   4. D) 6
3. Question 3: Which number is closest to $\sqrt{30}$?
   1. A) 4.5
   2. B) 5.0
   3. C) 5.5
   4. D) 6.0
4. Question 4: $\sqrt{50}$ is between which two whole numbers?
   1. A) 5 and 6
   2. B) 6 and 7
   3. C) 7 and 8
   4. D) 8 and 9
5. Question 5: Estimate $\sqrt{8}$ to the nearest tenth.
   1. A) 2.6
   2. B) 2.7
   3. C) 2.8
   4. D) 2.9
6. Question 6: Which value is the best estimate for $\sqrt{60}$?
   1. A) 7.2
   2. B) 7.5
   3. C) 7.7
   4. D) 7.8
7. Question 7: Which of the following is the closest estimation of $\sqrt{40}$?
   1. A) 5.8
   2. B) 6.1
   3. C) 6.3
   4. D) 6.7