Just as multiplication and division are opposites of each other (example: \( 3 \times 5 = 15 \) so \( 15 \div 5 = 3 \)), powers and roots are opposite. Because 5 raised to the 2 power = 25, the 2 root of 25 = 5. Roots can be significantly more difficult to find than powers because not every number has a simple root. To illustrate, \( 3^2 = 9 \) means the square root of 9 is 3. Similarly, \( 4^2 = 16 \) which means the square root of 16 is 4. But the numbers in between 9 and 16 don’t have a whole number square root because their roots must be somewhere between 3 and 4. Often we solve for roots using a calculator. The following video will help you learn how to solve for roots:

Real World Application

Some calculators have you type the number in first and then hit the square root button. Other calculators may have you do it the opposite way, by selecting the square root button and then typing in the number you want to root. You should experiment with your calculator on a simple square root, such as \(\sqrt{9}=3\), in order to see how your calculator works.

Video Source (05:44 mins) | Transcript

It’s helpful to learn which numbers are “perfect squares,” or the numbers that have whole number roots. These are the numbers that appear on the diagonal of a multiplication table because they are the result of any number being multiplied to itself. Some of these numbers include \(\text{4, 9, 16, 25, 36, 49, 64}\). We highly recommend that you memorize your multiplication facts to help you remember the perfect squares and their roots.

Additional Resources

• Khan Academy: Intro to Square Roots (05:23 mins, Transcript)
• Khan Academy: Understanding Square Roots (01:20 mins, Transcript)

Practice Problems

1. \(\sqrt{64}=\) (
   Solution

   x

   Solution: 8
   Details:
   The square root is looking for the number when multiplied to itself is equal to the number under the radical sign \(\sqrt{}\).
   
   It is highly suggested that you learn the multiplication facts. These will help you with questions like this one. According to our multiplication facts:
   
   \(8 \times 8 = 64\)
   
   Therefore, \(\sqrt{64} = 8\)
   )
2. \(\sqrt{49}=\) (
   Solution

   x

   Solution:
   7
   )
3. \(\sqrt{121}=\) (
   Video Solution

   x

   Solution:
   11
   Details:
   

   
   (Video Source | Transcript)
   )
4. \(\sqrt{7}=\) (
   Solution

   x

   Solution:
   \(2.65\) (Rounded to the nearest hundredth)
   Details:
   The square root is looking for the number when multiplied by itself is equal to the number under the radical sign \(\sqrt{}\).
   
   When a number is a perfect square, or the product of a whole number squared like \(3\times3=9\), then it is easier to find the square root. However, a number like 7 is not a perfect square. In cases like this, we can estimate the square root. It must be somewhere between 2 and 3 because \(2\times2=4\) and \(3\times3=9\) and 7 is between 4 and 9. In order to be more accurate though, we use a calculator.
   
   Your calculator may have a button that looks similar to one of these:
   
   
   
   Using a calculator, you should get the answer \(2.645751311\)…
   
   We can then round it to the precision needed for the question. In this case, we will round it to the nearest hundredth which is \(2.65\).
   )
5. \(\sqrt{78}=\) (
   Video Solution

   x

   Solution:
   \(8.83\) (Rounded to the nearest hundredth)
   Details:
   

   
   (Video Source | Transcript)
   )

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