Estimating square roots

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Intros
Lessons
    • What is a square root?
    • How to find side lengths of squares using square roots?
  1. How to estimate square roots?
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Examples
Lessons
  1. Estimate the values to two decimal places. Check the answers with a calculator.
    1. 24\sqrt {24}
    2. 66\sqrt {66}
    3. 110\sqrt {110}
  2. What is a whole number that has a square number between 8 and 9? Check your answer with a calculator.
    1. Alice has baked a square cake that has a top surface area of 335 cm2c{m^2}. She has a rectangular box that has a width of 20 cm and length of 27 cm.
      1. Estimate the side length of the cake to two decimal places. Check your answer with a calculator.
      2. Can the cake fit into the box? Why?
    Topic Notes
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    To estimate square roots which are not perfect squares without using a calculator, we need to know the perfect square numbers well. We will first put the number inside the square root sign in the middle of a number line, and then find the two closest perfect square numbers on its left and right hand side to make the best estimation.

    Square definition 

    Before we can understand what square roots are, we must first remember what squaring does. Squaring a number is to make a number multiply by itself. For example, 3^2 is equalled to 3 x 3, which gives us 9.

    Square root definition

     

    A square root is the inverse of squaring. What is an inverse? Inverses are opposites, so for example, the inverse of adding is subtracting. The inverse of multiplying is dividing. These are opposites of each other, and have an inverse relationship.

     

    What is a perfect square?

    When you take the square root of a number, you may get a decimal number. If you're able to get a whole number as an answer, then the original number you were finding the square root of is a perfect square number. Examples of a list of perfect squares include the numbers 4, 9, 16, 25, 36 and 49. Click here for a more comprehensive list of perfect squares, or check out this perfect square calculator.

     

    How to find square roots without calculators

    Let's move on to see some square root examples, and learn how to find answers for square roots. You'll first have to keep in mind the steps to finding square roots and they are:

    1. Estimate: Get as close as possible to the number you're trying to square root by finding two perfect square roots that gives a close number.
    2. Divide: Divide your number by one of the square roots you've chosen from the previous step.
    3. Average: Take the average of step 2 and the root.
    4. Repeat: Keep repeating steps 2 and 3 using the results you got from step 3 until you get a number that's accurate enough for you to answer the question.

     

    So in simpler words, to estimate square roots which are not perfect squares without using a calculator, we'll need to know the perfect square numbers well. We will first put the number inside the square root sign in the middle of a number line, and then find the two closest perfect square numbers on its left and right hand side to make the best estimation. Take a look at some of the below examples of square roots.

     

    Square root of 5

    Step 1: Estimate

    The square numbers of 2 and 3 are 4 and 9 respectively. The number 5 lies between these two numbers.

     

    Step 2: Divide

    Divide 5 by either 2 or 3. In this case, let's choose 2. We'll get 2.5.

    Step 3: Average

    Average 2.5 and 2, which gives us 2.25.

     

    Step 4: Repeat

    To get a more accurate number, keep repeating step 2 and 3. In that case we'd take 5 and divide it by 2.25, which equals 2.222. Average out 2.22 and 2.25, giving us 2.235. You may repeat steps 2 and 3 as many times needed to get a more accurate number.

     

    The final answer for the square root of 5 is approximately 2.23!

     

    Square root of 8

    Step 1: Estimate

    8 lies between perfect squares of 2^2 and 3^2.

     

    Step 2: Divide

    Divide 8 by 3. We get 2.6666666

     

    Step 3: Average

    Average 2.6666666 and 3, which gives us 2.8333333

     

    Step 4: Repeat

    To get a more accurate number, keep repeating step 2 and 3.

     

    You should get a final answer of 2.83.

     

    Square root of 10

    Step 1: Estimate

    10 lies between perfect squares of 3^2 and 4^2.

     

    Step 2: Divide

    Divide 10 by 3. We get 3.33

     

    Step 3: Average

    Average 3.33 and 3, which gives us 3.1667

     

    Step 4: Repeat

    To get a more accurate number, keep repeating step 2 and 3.

     

    You should get a final answer of 3.16.



    Square root of 6

    Step 1: Estimate

    6 lies between perfect squares of 2^2 and 3^2.

     

    Step 2: Divide

    Divide 6 by 2. We get 3.

     

    Step 3: Average

    Average 3 and 2, which gives us 2.5

     

    Step 4: Repeat

    To get a more accurate number, keep repeating step 2 and 3.

     

    You should get a final answer of 2.45.

     

    Square root of 12

    Step 1: Estimate

    12 lies between perfect squares of 3^2 and 4^2.

     

    Step 2: Divide

    Divide 12 by 4. We get 3.

     

    Step 3: Average

    Average 3 and 4, which gives us 3.5.

     

    Step 4: Repeat

    To get a more accurate number, keep repeating step 2 and 3.

     

    You should get a final answer of 3.46.

     

    Square root of 20

    Step 1: Estimate

    20 lies between perfect squares of 4^2 and 5^2.

     

    Step 2: Divide

    Divide 20 by 5. We get 4.

     

    Step 3: Average

    Average 4 and 5, which gives us 4.5.

     

    Step 4: Repeat

    To get a more accurate number, keep repeating step 2 and 3.

     

    You should get a final answer of 4.47.

     

    Square root of 0

    As a final note, we wanted to explore what the square root of 0 is. You can't take the square root of a negative number, but 0 is not a negative number. The square root of 0 is actually 0!

     

    If you wanted to try finding more square roots, try out this fun game with lots of examples.