If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Square root of decimal

Learn how to find the square root of a decimal number. The problem solved in this video is p^2 = 0.81.

Want to join the conversation?

  • ohnoes default style avatar for user Lloyd Christmas
    What if the decimal doesn't go in as easily as .9 to .81?
    (66 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Woohoo
      Use a calculator, but if you can't you could do this:
      p^2=0.50
      p=√0.50
      p=√0.50
      p=√0.25 *√2
      p=0.5*√2
      p=1/2 *√2
      p=√2/2
      It's just a simple example, but it works, and it can help you with some of the more complicated problems.
      If you're wondering about the problems like the square root of 0.15, well, those cant be simplified because they don't have square roots inside of them (if you don't believe me, look it up or check your calculator)
      If you did not notice, √0.50 has 0.25, which is a square root (0.5^2 = 0.25, √0.25=0.5). So if you do this and can't find the square roots inside of the root you're solving for, that just means it can't be simplified or solved for, so don't panic. Just look for square roots inside the root you're solving for, and if you don't find any, that means it can't be simplified.
      If you want to go further:
      First, you would find the √2:
      You can use the Estimation and Approximation Method:
      y=√x
      ((x/y)+y)/2
      y_n+1 = ((x/y_n) + y_n)/2
      y1 = (2 + 1)/2 = 1.5
      y2 = (4/3 + 3/2)/2 = 1.4166
      y3 = (24/17 + 17/12)/2 = 1.414215...
      So, then you would divide that by two (I will be going to the thousands place)~
      1.414/2 =(aproximatly) 0.707
      So √2/2 is about 0.707.
      (In the comments there is a link to the website I used for this, it won't allow me to put it in)
      ~ Woohoo
      (32 votes)
  • aqualine ultimate style avatar for user Purajit Ghosh
    Isn't -0.9 squared equal to -0.81. Technically, (-0.9) squared is the answer. Use the calculator. I think I am right. Am I?
    (13 votes)
    Default Khan Academy avatar avatar for user
  • marcimus pink style avatar for user sofia.fuentes
    at why does he put the plus or minus sign?
    (10 votes)
    Default Khan Academy avatar avatar for user
  • winston default style avatar for user Anderson Le
    What if the number was a fraction? Would I try to just simplify it down to shortest terms possible in decimal form and work from there? I have a problem of finding the square root of 49/81.
    (22 votes)
    Default Khan Academy avatar avatar for user
  • starky sapling style avatar for user Ian Pape
    what about number like the square root of 1.69 or 4.84?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • stelly blue style avatar for user Kim Seidel
      Find the square root of 169 (no decimal points). It = 13.
      Then figure out how many decimal points you need in your answer. 1.3 x1.3 = 1.69.
      So, sqrt(1.69) = 1.3

      Alternatively, do them as fractions. 1.69 = 169/100
      sqrt(169/100) = sqrt(169) / sqrt(100) = 13/10 = 1.3
      (29 votes)
  • leaf green style avatar for user Aidan Jalili
    At Sal counts the number of decimal points in the expression, in order to find the number of decimal points in the answer. I've seen this been done before, and I'm familiar with using the technique, but I'm wondering why it works. I've simply taken my teachers word and simply assumed it works because my teacher said so. But now I'm actually curious, why does this technique, for finding the number of decimal places in the answer, work?
    (15 votes)
    Default Khan Academy avatar avatar for user
    • female robot ada style avatar for user Angelica Chen
      You can understand it by switching to scientific notation. A number like a•10^3 • b•10^4 is solved by adding the exponents on the tens, showing that when you multiply two numbers, the number of digits in them add up (or are at most one off).
      (1 vote)
  • blobby green style avatar for user Emma Barella
    What if the decimal does not go in as easy as 0.9 to 0.81?
    (4 votes)
    Default Khan Academy avatar avatar for user
    • mr pink green style avatar for user David Severin
      Since we break square roots into pairs of numbers, it would be no different than trying to find the square root of 3 or 5 or 41. These would equate to .03, .05, or .41. If it is a perfect square whole number, then two places would make it a perfect square decimal. (.01, .04, .09, .16, .25, .36, .49, .64, .81, or even .0121, .0144, etc.). All other decimals form the same struggle as whole numbers. The reason is that .81 = 81/100 and 100 is always a perfect square, so you get 9/10=.9. If you try .55, that gives 55/100 which you can take square root of denominator, but not the numerator 55.
      (4 votes)
  • piceratops tree style avatar for user Nandana
    how to calculate the cube root of decimal numbers?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • purple pi purple style avatar for user wazzu
      In the "into to cube roots" video Sal showed how doing the prime factorization of a number can help you figure out its cube root. Take 64 which can break down into 4x16; 16 breaks down into 4x4; so if you multiply 4x4x4 you get 64. Take 0.125 which can break down into .5x.25; .25 breaks down into .5x.5; so if you multiply .5x.5x.5 you get .125
      (7 votes)
  • duskpin sapling style avatar for user Elizabeth
    What is the difference between decimals and negative numbers?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • female robot grace style avatar for user Evelyn Davis
      A negative number is less then 1, so like -1 or even -19. If you owed some one 5 dollars, and had no other money, your balance would be -5. A decimal is less than one as well, but in a different way. It is still a real number. It is similar to fractions, because it is a part of a whole. Half a cookie (.5) is very different than owing someone a cookie (-1).
      (2 votes)
  • hopper cool style avatar for user Muhammad Abdul
    What if the number was a fraction? Would I try to just simplify it down to shortest terms possible in decimal form and work from there? I have a problem of finding the square root of 49/81
    (3 votes)
    Default Khan Academy avatar avatar for user
    • piceratops seed style avatar for user Chic- Fila
      Well, for help on finding the square root of your fraction. I know that 7 x 7 is 49. And that 81 is 9 x 9. So then you would put 7/9 as your answer. Knowing at least your 12 x 12 times tables really helps with these things! But yes, generally you would want to simply your answer.

      I hope I was able to help! <3
      (4 votes)

Video transcript

- Let's see if we can solve the equation P squared is equal to 0.81. So how could we think about this? Well one thing we could do is we could say, look if P squared is equal to 0.81, another way of expressing this is, that well, that means that P is going to be equal to the positive or negative square root of 0.81. Remember if we just wrote the square root symbol here, that means the principal root, or just the positive square root. But here P could be positive or negative, because if you square it, if you square even a negative number, you're still going to get a positive value. So we could write that P is equal to the plus or minus square root of 0.81, which kind of helps us, it's another way of expressing the same, the same, equation. But still, what could P be? In your brain, you might immediately say, well okay, you know if this was P squared is equal to 81, I kinda know what's going on. Because I know that nine times nine is equal to 81. Or we could write that nine squared is equal to 81, or we could write that nine is equal to the principal root of 81. These are all, I guess, saying the same truth about the universe, but what about 0.81? Well 0.81 has two digits behind, to the right of the decimal and so if I were to multiply something that has one digit to the right of the decimal times itself, I'm gonna have something with two digits to the right of the decimal. And so what happens if I take, instead of nine squared, what happens if I take 0.9 squared? Let me try that out. Zero, I'm gonna use a different color. So let's say I took 0.9 squared. 0.9 squared, well that's going to be 0.9 times 0.9, which is going to be equal to? Well nine times nine is 81, and I have one, two, numbers to the right of the decimal, so I'm gonna have two numbers to the right of the decimal in the product. So one, two. So that indeed is equal to 0.81. In fact we could write 0.81 as 0.9 squared. So we could write this, we could write that P is equal to the plus or minus, the square root of, instead of writing 0.81, I could write that as 0.9 squared. In fact I could also write that as negative 0.9 squared. Cause if you put a negative here and a negative here, it's still not going to change the value. A negative times a negative is going to be a positive. I could, actually I would have put a negative there, which would have implied a negative here and a negative there. So either of those are going to be true. But it's going to work out for us because we are taking the positive and negative square root. So this is going to be, P is going to be equal to plus or minus 0.9. Plus or minus 0.9, or we could write it that P is equal to 0.9, or P could be equal to negative 0.9. And you can verify that, you would square either of these things, you get 0.81.