Jennifer has an MS in Chemistry and a BS in Biological Sciences.
The Language of Math
Mathematics has its own special language. To understand and succeed in math, you need to be able to translate the language of math into English and back again. Just like you wouldn't move to France without knowing how to communicate at least the basics, you shouldn't venture into the study of math without knowing how to translate the terms you are using.
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Squares and Square Roots
The section of math dealing with radicals , or squares and square roots, is one place where knowing the translations of the terms is critical to understanding.
To square a number means to multiply that number by itself. It is notated by a superscript number 2 after the main number. It can also be written with a caret between the number being squared and the 2. When you see the carat symbol (^), you say that the number is squared.
3^2 is equal to 3*3 or 9
10^2 is equal to 10*10 or 100
The square root operation is the inverse of the squared function and is notated by this symbol:
This means that finding the square root of a number is the same as finding the opposite of a number squared.
√100 is 10 because 10^2 is 100
√ 9 is 3 because 3^2 is 9
A number is a perfect square if it is an integer that is the square of another integer. An integer is a number that does not contain a fraction or a decimal.
5^2 is 25
So, 25 is a perfect square.
17^2 is 289, which means 289 is also a perfect square.
Finding the Square Root of a Perfect Square
You can use a calculator to find the square root of a perfect square, but, if a calculator is not available, there is a way to calculate the square root.
Here are the steps to calculating the square root of a perfect square:
Step one: factor the number completely. An easy way to factor a number is by using a factor tree. A factor tree can be created by writing down the number you want to factor and drawing two lines coming down from that number. Then, write two factors of that number under the lines. Continue on until only prime numbers remain. A prime number is one that cannot be reduced any smaller. The purpose of the factor tree is so you can easily find the square root of large numbers if you don't have a calculator handy.
By looking at the factor tree below, you can see that the factors of 225 are 3 * 3 * 5 * 5.
Step number 2 is to match up pairs of factors. In this case, we have a pair of 3s and a pair of 5s:
(3 * 3) * (5 * 5)
Then, step number 3 to calculating the square root of a perfect square is to multiply one number from each pair of factors together to get the answer. In this case:
3 * 5 = 15
Fifteen is the square root of 225.
For example, let's find the square root of 144. Your first step is to factor 144. The prime factors of 144 are (2*2) * (2*2) * (3*3)
Your second step is to match up the pairs. There are 2 pairs of 2s, and 1 pair of 3s.
Then, for step 3, take out 1 number from each pair, and multiply them together: 2*2*3, which equals 12.
Therefore, the square root of 144 is equal to 12.
Let's try another one. Solve √400.
Again, your first step is to factor 400 by using a factor tree. The result you get is (2*2) * (2*2) * (5*5).
Secondly, match up pairs of numbers. There are 2 pairs of 2s and 1 pair of 5s. Finally, multiply together the 2*2*5 to get 20.
So, the √400 = 20
Squares and square roots are inverse operations. A perfect square is the square of a number that does not contain a fraction or decimal. To determine the square root of a perfect square, all you need to do is factor the number, combine pairs of factors and then multiply one number from each pair together. You can always check your work by performing the inverse operation, or squaring the number to see if your answer matches up with the original square root.
After you have finished with this lesson, you should be able to:
- Describe the relationship between squares and square roots
- Define perfect square
- Explain how to determine the square root of a perfect square
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Evaluating Square Roots of Perfect Squares
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