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 roots of perfect squares

Learn how to find the square root of perfect squares like 25, 36, and 81.
Let's start by taking a look at an example evaluating the square root of 25:
25=?
Step 1: Ask, "What number squared equals 25?"
Step 2: Notice that 5 squared equals 25.
52=5×5=25
The answer
25=5
Here's a question to make sure you understood:
How can we be sure that 5 is the right answer?
Choose 1 answer:

Connection to a square

Finding the square root of 25 is the same as finding the side length of a square with an area of 25.
A square with unknown side lengths. The area is equal to 25.
A square with an area of 25 has a side length of 5.
A square with side legs of 5 units. The area is equal to 5 times 5, which is equal to 25.
A factor tree for 25. The number 25 is the product of 5 and 5. One of the number 5's is circled and the other is not.

Practice Set 1:

Problem 1A
42=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Reflection question

Which claim shows how square roots work?
Choose 1 answer:

Practice Set 2:

Problem 2A
1=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Practice Set 3:

Problem 3A
121=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Want to join the conversation?

  • blobby green style avatar for user cyanne.13.edwards
    Can an exponent be a negative number?
    (82 votes)
    Default Khan Academy avatar avatar for user
    • duskpin seedling style avatar for user Joann
      You already know that an exponent represents the number of times you have to multiply a number by itself. A negative exponent is equivalent to the inverse of the same number with a positive exponent. There is nothing special about solving a problem that includes negative exponentials.
      (27 votes)
  • male robot donald style avatar for user Saaketh Gaddam
    What happens if try to find the
    the square root of an imperfect square
    (16 votes)
    Default Khan Academy avatar avatar for user
    • cacteye blue style avatar for user Jerry Nilsson
      Most probably it will be an irrational number in which case we can only approximate its value.

      However, sometimes we can write the radicand as a fraction of two perfect squares.
      Example: √5.76 = √(576∕100) = √(144∕25) = √(12²∕5²) = 12∕5 = 2.4
      (34 votes)
  • mr pink red style avatar for user minnieli1223
    I just noticed something really interesting (I think):

    If I can't remember a square of some numbers (7^2 and 8^2 can be a bit tricky for me for some reason) but I remember the square number of the root that comes before it (6 and 36 in the case I'm trying to find 7^2),
    I can do 36+6 to make it into 6*7, and then add a 7 to make it into a 7*7.

    I tried to play around and find a rule and I think I found the formula:
    n^2 + 2n + 1 = ( n + 1 )^2
    (where n is the root number of the square that you do know).

    if you would visualize the numbers on a grid, the n^2 is the area of the square, while the 2n+1 is the number of the additional units the is added on the side.

    It's easier to see it on a times table: When looking at 25, the number diagonally next to it is 36. If you count the 'units' (the other multiplies that are on the same axes that leads toward 36 IE: 3,6,9 up to 36 on both sides) they will be the same as the 2n+1.

    I tried to find a formula for a square of a root that isn't immediately follows the root I know. example: 3^2 = 9, 5^2 = ?.
    unfortunately I couldn't think on one consistent formula, because there is always a need to add more and (n+1) with more additional 1s the further the number is.

    does the formula I found have a name? I'm pretty sure I wasn't the first to think of that lol
    (15 votes)
    Default Khan Academy avatar avatar for user
  • hopper cool style avatar for user maruandtotoro
    So, a perfect square is basically the answer to an exponent? (ex. 2^2(4), 12^12(144))
    (9 votes)
    Default Khan Academy avatar avatar for user
  • leafers seedling style avatar for user jl974294
    How do you solve square roots that can't be squared easily?
    (8 votes)
    Default Khan Academy avatar avatar for user
  • female robot ada style avatar for user aniza white
    How can we use square roots in life?
    (7 votes)
    Default Khan Academy avatar avatar for user
  • starky sapling style avatar for user Gigi💙
    Would you do the same for a negative square root ?🤔
    (3 votes)
    Default Khan Academy avatar avatar for user
    • leaf orange style avatar for user Benny C
      You cannot find the negative square root of a number. Think of what square roots are. If we take 36 and find the square root, you'll see it's 6. Because 6 x 6 = 36. We multiply it by itself and it gives us 36.

      We can't do this with negatives. Consider -36 and finding the square root of that. Which number multiplied by itself will get -36?

      Well, remember the rules we have for multiplying negatives and positives.
      negative x negative = positive
      negative x positive = positive
      positive x positive = positive.

      It can't be -6, because that will give us a positive number, instead of -36.
      It can't be 6, because that will also give us a positive number instead of -36.

      So, negatives can't have square roots.

      You'll learn in more advanced classes what we do in these cases.
      (10 votes)
  • starky seedling style avatar for user Suryansh Singh
    Math is hard!
    (5 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user rebecca.ndorimana.28
    How do I find the square root for bigger and not perfect numbers?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • male robot donald style avatar for user Venkata
      There are a couple of ways:

      1. You can approximate the value. For instance, if I have $\sqrt{70}$, I know that it'll lie between $\sqrt{64}$ and $\sqrt{81}$. As 70 is closer to 64, $\sqrt{70}$ will be closer to $\sqrt{64}$, which is 8. And if you find $\sqrt{70}$, you'll see that it is around $8.366$, which is closer to 8 than 9.

      2. You can approximate the square root function with a tangent line at a nearby point. Suppose you want $\sqrt{65}$. You already know $\sqrt{64}$. So, you can first graph $y=\sqrt{x}$, then draw a tangent at $x = 64$. Find the equation of the tangent, and substitute for $x = 65$. As the tangent approximates the square root function, you'll get an rough approximation of $\sqrt{65}$.

      Using this, you can get the equation of the tangent line as $y = \frac{x}{16}+4$. If I substitute $x = 65$, I get $y = 8.0625$. Now, the actual value of $\sqrt{65}$ is $8.06225$. See how good our approximation was!

      3. The final method is something called a Taylor series. It's an extension of the tangent line approximation idea. If you write the Taylor series expansion of $\sqrt{x}$ at $x = 64$ to four terms, you get $8 + \frac{x-64}{16} - \frac{(x-64)^2}{4096} + \frac{(x-64)^3}{524288}$. If I substitute $x = 65$, I'll get $8 + \frac{1}{16} - \frac{1}{4096} + \frac{1}{524288} = 8.062257$, which is an even better approximation than what we got in method 2. It gets better with more terms.

      I know you aren't familiar with methods 2 and 3. But hey, if you stick with Math long enough, you'll one day come back to this comment and understand everything!
      (7 votes)
  • starky tree style avatar for user Daniel Rodgers
    It's actually a lot easier than I thought
    (7 votes)
    Default Khan Academy avatar avatar for user