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Approximating square roots to hundredths

Let's approximate the square root of 45 without a calculator. We'll explore how to find the perfect squares around 45 and use them to make an educated guess. Then, we'll refine our guess by squaring it to see how close we get to 45. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • aqualine ultimate style avatar for user Riya
    What are the differences between square roots and cube roots?
    (143 votes)
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  • leafers ultimate style avatar for user John Jairo Amaya
    what is a perfect square and what does it mean?
    (32 votes)
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    • piceratops tree style avatar for user Raeda Sarwar
      A perfect square is a number that can be expressed as the product of two equal integers.

      Examples of perfect squares:
      * 9
      o 9 is a perfect square becuase it can be expressed as 3 * 3 (the product of two equal integers)
      * 16
      o 16 is a perfect square becuase it can be expressed as 4 * 4 (the product of two equal integers)
      * 25
      o 25 is a perfect square becuase it can be expressed as 5 * 5 (the product of two equal integers)

      NON examples of perfect squares:

      … (more) * 8
      o 8 is a not perfect square because you cannot express it as the product of two equal integers
      * 5
      o 5 is a not perfect square because it cannot be expressed as the product of two equal integers
      * 7
      o 7 is a not perfect square because you cannot express it as the product of two equal integers

      hope this helps :)
      (51 votes)
  • duskpin ultimate style avatar for user Maruschka
    at what does he mean by "nine of the way through it? Please help
    (17 votes)
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  • starky sapling style avatar for user KI5YNW
    I found this confusing a bit. Did anyone else?
    (12 votes)
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  • blobby green style avatar for user alexmirzabdullaev
    why not just use Desmos calculator
    (6 votes)
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    • blobby green style avatar for user Lucky
      probably because if they allowed calculators people could just solve it with a built-in square root function and not have any actual knowledge of how it works (like what you seem to be suggesting), also, things like Desmos (and really, modern technology in general) only exist because people did math to find accurate and efficient ways to calculate things like this
      (11 votes)
  • leafers ultimate style avatar for user ben
    What is the difference between square roots and cube roots
    (2 votes)
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    • hopper cool style avatar for user oriramikad
      Squaring a number multiplies twice. Some squared numbers:
      1² = 1 * 1
      2² = 2 * 2
      3² = 3 * 3
      4² = 4 * 4
      5² = 5 * 5
      Cubing a number multiplies three times. Some cubed numbers:
      1³ = 1 * 1 * 1
      2³ = 2 * 2 * 2
      3³ = 3 * 3 * 3
      4³ = 4 * 4 * 4
      5³ = 5 * 5 * 5
      And so on.
      But when we take the ROOT of a number, what we are actually doing is asking a question. When we get the square root of a number we are asking, "What number times what number equals the number we are squaring?" For example:
      √4 = 2
      The square root of 4 equals 2. Why? Because 2 times 2 equals 4. Another example:
      √9 = 3
      The square root of 9 equals 3. Why? Because 3 times 3 equals 9.
      Now, the difference between square roots and cube roots is that with cube roots, we are asking a similar question, but the amount that the numbers need to multiply changes.
      ³√8 = 2
      The cube root of 8 equals 2. Why? Because 2 times 2 times 2 equals 8. Another example:
      ³√27 = 3
      The cube root of 27 equals 3. Why? Because 3 times 3 times 3 equals 27.
      I hope you were able to understand and get through all that! It was a rather hefty manuscript. :)
      Toodleoo! *tips hat*
      (15 votes)
  • blobby green style avatar for user STEWIE
    i still don't understand...
    (8 votes)
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  • blobby green style avatar for user tholer763
    How does he get 9/13
    (6 votes)
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  • starky sapling style avatar for user leeanne
    how did he get the 9 in 9/13?
    (6 votes)
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    • duskpin ultimate style avatar for user ✟lilac pineapple✟
      Hello!

      Do you understand the part where Sal says that 45 (and the square root of 45) is halfway between 49 (and the square root of 49) and 36 (and the square root of 36)?

      At , Sal explains that the difference between 49 and 45 is 4, and that the difference between 45 and 36 is 9. Therefore, the difference between 49 and 36 is 13.

      He then puts the 9 above the 13 to get 9/13!

      This was confusing for me too, but I hope this helped!
      (8 votes)
  • blobby green style avatar for user omar
    Wouldn't 6 + 9/13 be the square root of 45?
    (2 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      This is an interesting question. It is true that 45 is 9/13 of the way from 36 to 49. However, because the square root function is a nonlinear function, the previous sentence does not mean that sqrt(45) is exactly 9/13 of the way from sqrt(36)=6 to sqrt(49)=7.

      Since the graph of the square root function is concave down, sqrt(45) is larger than 6 + 9/13; sqrt(45) is about 6.708, but 6 + 9/13 is only about 6.692.
      Visually, the graph of the function y=sqrt(x) on the interval 36<x<49 is above the graph of the line segment joining (36, 6) and (49, 7) but excluding the endpoints. This is why it makes sense for sqrt(45) to exceed 6 + 9/13. So 6 + 9/13 is only an approximation for sqrt(45).

      Have a blessed, wonderful day!
      (12 votes)

Video transcript

We are asked to approximate the principal root, or the positive square root of 45, to the hundredths place. And I'm assuming they don't want us to use a calculator. Because that would be too easy. So, let's see if we can approximate this just with our pen and paper right over here. So the square root of 45, or the principal root of 45. 45 is not a perfect square. It's definitely not a perfect square. Let's see, what are the perfect squares around it? We know that it is going to be less than-- the next perfect square above 45 is going to be 49 because that is 7 times 7-- so it's less than the square root of 49 and it's greater than the square root of 36. And so, the square root of 36, the principal root of 36 I should say, is 6. And the principal root of 49 is 7. So, this value right over here is going to be between 6 and 7. And if we look at it, it's only four away from 49. And it's nine away from 36. So, the different between 36 and 49 is 13. So, it's a total 13 gap between the 6 squared and 7 squared. And this is nine of the way through it. So, just as a kind of approximation maybe-- and it's not going to work out perfectly because we're squaring it, this isn't a linear relationship-- but it's going to be closer to 7 than it's going to be to 6. At least the 45 is 9/13 of the way. Let's see. It looks like that's about 2/3 of the way. So, let's try 6.7 as a guess just based on 0.7 is about 2/3. It looks like about the same. Actually, we could calculate this right here if we want. Actually, let's do that just for fun. So 9/13 as a decimal is going to be what? It's going to be 13 into 9. We're going to put some decimal places right over here. 13 doesn't go into 9 but 13 does go into 90. And it goes into 90-- let's see, does it go into it seven times-- it goes into it six times. So, 6 times 3 is 18. 6 times 1 is 6, plus 1 is 7. And then you subtract, you get 12. So, went into it almost exactly seven times. So, this value right here is almost a 0.7. And so if you say, how many times does 13 go into 120? It looks like it's like nine times? Yeah, it would go into it nine times. 9 times 3. Get rid of this. 9 times 3 is 27. 9 times 1 is 9, plus 2 is 11. You have a remainder of 3. It's about 0.69. So 6.7 would be a pretty good guess. This is 0.69 of the way between 36 and 49. So, let's go roughly 0.69 of the way between 6 and 7. So this is once again just to approximate. It's not necessarily going to give us the exact answer. We have to use that to make a good initial guess. And then see how it works. Let's try 6.7. And the really way to try it is to square 6.7. So 6.7 times-- maybe I'll write the multiplication symbol there-- 6.7 times 6.7. So, we have 7 times 7 is 49. 7 times 6 is 42, plus 4 is 46. Put a 0 now because we've moved a space to the left. So, now we have 6 times 7 is 42. Carry the 4. 6 times 6 is 36, plus 4 is 40. And so, 9 plus 0 is 9. 6 plus 2 is 8. 4 plus 0 is 4. And then we have a 4 right over here. And we have two total numbers behind the decimal point. One, two. So this gives us 44.89. So, 6.7 gets us pretty close. But we're still not probably right to the hundredth. Well, we're definitely not to the hundredths place. This since we've only gone to the tenths place right over here. So, if we want to get to 45, 6.7 squared is still less than 45, or 6.7 is still less than the square root of 45. So let's try 6.71. Let me do this in a new color. I'll do 6.71 in pink. So, let's try 6.71. Increase it a little bit. See if we go from 44.89 to 45. Because this is really close already. Let's just try it out. 6.71. So once again, we have to do some arithmetic by hand. We are assuming that they don't want us to use a calculator here. So, we have 1 times 1 is 1. 1 times 7 is 7. 1 times 6 is 6. Put a 0 here. 7 times 1 is 7. 7 times 7 is 49. 7 times 6 is 42, plus 4 is 46. And then we have two 0s here. 6 times 1 is 6. 6 times 7 is 42. Just have this new 4 here. 6 times 6 is 36, plus 4 is 40. Plus 40. It's interesting to think what we got incrementally by adding that one hundredth over there. Well, we'll see actually when we add all of this up. You get a 1. 7 plus 7 is 14. 1 plus 6 plus 9 is 16, plus 6 is 22. 2 plus 6 plus 2 is 10. And then 1 plus 4 is 5. And then we bring down the 4. And we have one, two, three, four numbers behind the decimal point. One, two, three, four. So, when you we squared 6.71. 6.71 squared is equal to 45.0241. So 6.71 is a little bit greater. So, let me make it clear now. We know that 6.7 is less than the square root of 45. And we know that is less than 6.71. Because when we square this, we get something a little bit over the square root of 45. But the key here is when we square this, so 6.7 squared got us 44.89 which is 0.11 away from 45. And then, if we look at 6.71 squared, we're only 2.4 hundredths above 45. So, this right here is closer to the square root of 45. So if we approximate to the hundredths place, definitely want to go with 6.71.