Estimate positive and negative square roots

Key Notes :

1. Understanding Square Roots:

  • The square root of a number is a value that, when multiplied by itself, gives the original number.
  • Square roots of positive numbers have two values: a positive and a negative root.
    • Example: √16 = 4 and -4 because 4 × 4 = 16 and (-4) × (-4) = 16.

2. Estimating Square Roots:

  • When square roots are not perfect squares (like √8 or √20), they need to be estimated.
  • Use nearest perfect squares to find a range for the square root:
    • Example: To estimate √8, recognize that 4² = 16 and 3² = 9. So, √8 is between 2 and 3.
  • For better accuracy, estimate between these values by identifying closer perfect squares.

3. Estimating Positive Square Roots:

  • For square roots between two perfect squares, use trial and error or a calculator for better precision.
  • Example: To estimate √18, note that 4² = 16 and 5² = 25. Thus, √18 is between 4 and 5.
    • Check closer values: Try √18 ≈ 4.24 (since 4.24 × 4.24 ≈ 18).

4. Estimating Negative Square Roots:

  • The negative square root of a number is simply the negative of the positive square root.
  • Example: For √9 = 3, the negative square root is -3.
  • For non-perfect squares, you can similarly estimate the negative square root by applying the same method.
    • Example: To estimate √7 ≈ 2.65, the negative square root is -2.65.

5. Using a Number Line:

  • Drawing a number line can help visualize square roots and estimate their values.
  • Mark the closest perfect squares on the number line to help locate the estimated square root.

6. Estimation Process:

  1. Find the nearest perfect squares.
  2. Identify the range between the two square roots.
  3. Use trial and error or a calculator for more precise estimates.

7. Examples:

Estimate √17:

  • Closest perfect squares are 16 (4²) and 25 (5²). So √17 is between 4 and 5.
  • Use a calculator for more precision: √17 ≈ 4.123.

Estimate √50:

  • Closest perfect squares are 49 (7²) and 64 (8²). So √50 is between 7 and 8.
  • Use trial values: √50 ≈ 7.071.

Learn with an example

Complete the following statement. Use the integers that are closest to the number in the middle.

( ) <-√15< ( )

For now, ignore the negative sign. Find the perfect squares that are just below and just above 15.

The perfect square just below 15 is 9:

√9 = 3

The perfect square just above 15 is 16:

√16 = 4

√15 is between √9 and √16 , so √15 is between 3 and 4 .

Finally, include the negative signs. Remember that with negative numbers, larger numbers like -√16 (if you ignore the minus sign) are less than smaller numbers.

-√16< – √15 <-√9 , so -4<-√15<-3

Complete the following statement. Use the integers that are closest to the number in the middle.

( ) <-√7< ( )

For now, ignore the negative sign. Find the perfect squares that are just below and just above 7.

The perfect square just below 7 is 4:

√4 = 2

The perfect square just above 7 is 9:

√9 = 3

√7 is between √4 and √9 , so √7 is between 2 and 3.

Finally, include the negative signs. Remember that with negative numbers, larger numbers like -√9 (if you ignore the minus sign) are less than smaller numbers.

-√9<-√7<-√4 , so -3<-√7<-2 .

Let’s try some examples! ✍️